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10.9: Right Triangle Trigonometry

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    129647
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    Pythagoras is shown writing in a book as a young man presents him with a tablet showing a diagrammatic representation of a lyre above a drawing of the sacred tetractys.
    Figure 10.178 In the lower left corner of the fresco The School of Athens by Raphael, the figure in white writing in the book represents Pythagoras. Alongside him, to the right, the figure with the long, light-brown hair is said to depict Archimedes. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, Public Domain)

    Learning Objectives

    After completing this section, you should be able to:

    1. Apply the Pythagorean Theorem to find the missing sides of a right triangle.
    2. Apply the 30-60-9030-60-90 and 45-45-9045-45-90 right triangle relationships to find the missing sides of a triangle.
    3. Apply trigonometric ratios to find missing parts of a right triangle.
    4. Solve application problems involving trigonometric ratios.

    This is another excerpt from Raphael’s The School of Athens. The man writing in the book represents Pythagoras, the namesake of one of the most widely used formulas in geometry, engineering, architecture, and many other fields, the Pythagorean Theorem. However, there is evidence that the theorem was known as early as 1900–1100 BC by the Babylonians. The Pythagorean Theorem is a formula used for finding the lengths of the sides of right triangles.

    Born in Greece, Pythagoras lived from 569–500 BC. He initiated a cult-like group called the Pythagoreans, which was a secret society composed of mathematicians, philosophers, and musicians. Pythagoras believed that everything in the world could be explained through numbers. Besides the Pythagorean Theorem, Pythagoras and his followers are credited with the discovery of irrational numbers, the musical scale, the relationship between music and mathematics, and many other concepts that left an immeasurable influence on future mathematicians and scientists.

    The focus of this section is on right triangles. We will look at how the Pythagorean Theorem is used to find the unknown sides of a right triangle, and we will also study the special triangles, those with set ratios between the lengths of sides. By ratios we mean the relationship of one side to another side. When you think about ratios, you should think about fractions. A fraction is a ratio, the ratio of the numerator to the denominator. Finally, we will preview trigonometry. We will learn about the basic trigonometric functions, sine, cosine and tangent, and how they are used to find not only unknown sides but unknown angles, as well, with little information.

    Pythagorean Theorem

    The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle).

    FORMULA

    The Pythagorean Theorem states

    a2+b2=c2a2+b2=c2

    where aFigure 10.179.

    A right triangle with its legs marked b and a. The hypotenuse is marked c.
    Figure 10.179 Pythagorean Right Triangle

    For example, given that side a=6,a=6, and side b=8,b=8, we can find the measure of side cc using the Pythagorean Theorem. Thus,

    a2+b2=c2(6)2+(8)2=c236+64=c2100=c2100=c210=ca2+b2=c2(6)2+(8)2=c236+64=c2100=c2100=c210=c

    Example 10.64

    Using the Pythagorean Theorem

    Find the length of the missing side of the triangle (Figure 10.180).

    A right triangle with its legs marked 6 and b. The hypotenuse is marked 14.
    Figure 10.180
    Answer

    Using the Pythagorean Theorem, we have

    ( 6 ) 2 + b 2 = ( 14 ) 2 36 + b 2 = 196 b 2 = 196 36 b 2 = 160 b = ± 160 = 4 10 = 12.65 ( 6 ) 2 + b 2 = ( 14 ) 2 36 + b 2 = 196 b 2 = 196 36 b 2 = 160 b = ± 160 = 4 10 = 12.65

    When we take the square root of a number, the answer is usually both the positive and negative root. However, lengths cannot be negative, which is why we only consider the positive root.

    Your Turn 10.64

    1.
    Use the Pythagorean Theorem to find the missing side of the right triangle shown.
    A right triangle with its legs marked 12 and a. The hypotenuse is marked 13.
    Figure 10.181

    Distance

    The applications of the Pythagorean Theorem are countless, but one especially useful application is that of distance. In fact, the distance formula stems directly from the theorem. It works like this:

    In Figure 10.182, the problem is to find the distance between the points (3,1)(3,1) and (3,2).(3,2). We call the length from point (3,1)(3,1) to point (3,1)(3,1) side aa, and the length from point (3,1)(3,1) to point (3,2)(3,2) side bb. To find side cc, we use the distance formula and we will explain it relative to the Pythagorean Theorem. The distance formula is d=(x2x1)2+(y2y1)2,d=(x2x1)2+(y2y1)2, such that (x2x1)(x2x1) is a substitute for aa in the Pythagorean Theorem and is equal to 3(3)=6;3(3)=6; and (y2y1)(y2y1) is a substitute for bb in the Pythagorean Theorem and is equal to 2(1)=3.2(1)=3. When we plug in these numbers to the distance formula, we have

    d=(3(3))2+(2(1))2=(6)2+(3)2=36+9=45=35=6.7d=(3(3))2+(2(1))2=(6)2+(3)2=36+9=45=35=6.7

    Thus, d=cd=c, the hypotenuse, in the Pythagorean Theorem.

    A right triangle is plotted on an x y coordinate grid. The vertices of the triangle are (negative 3, negative 1), (3, negative 1), and (3, 2). The distance from (negative 3, negative 1) to (3, 2) is labeled c. The distance from (negative 3, negative 1) to (3, negative 1) is labeled a. The distance from (3, negative 1) to (3, 2) is labeled b.
    Figure 10.182 Distance

    Example 10.65

    Calculating Distance Using the Distance Formula

    You live on the corner of First Street and Maple Avenue, and work at Star Enterprises on Tenth Street and Elm Drive (Figure 10.183). You want to calculate how far you walk to work every day and how it compares to the actual distance (as the crow flies). Each block measures 200 ft by 200 ft.

    A right triangle is plotted on a rectangular grid. The horizontal leg shows a star on the left and it is labeled 0. The right end of the leg is labeled 1800. This leg represents Elm Drive. The vertical leg represents First Street. The bottom and top ends of the leg represent 0 and 1400. A house is near 1400. A dashed line representing the hypotenuse connects star and house. Maple Avenue is along the first row of the grid. Tenth Street is along the first column of the grid.
    Figure 10.183
    Answer

    You travel 7 blocks south and 9 blocks west. If each block measures 200 ft by 200 ft, then 9(200)+7(200)=1,800ft+1,400ft=3,200ft9(200)+7(200)=1,800ft+1,400ft=3,200ft.

    As the crow flies, use the distance formula. We have

    d = ( 1,800 0 ) 2 + ( 1,400 0 ) 2 = 3,240,000 + 1,960,000 = 5,200,000 = 2280.4 ft d = ( 1,800 0 ) 2 + ( 1,400 0 ) 2 = 3,240,000 + 1,960,000 = 5,200,000 = 2280.4 ft

    Your Turn 10.65

    1.
    How far is it to your workplace (as the crow flies) if the blocks in the previous example measure 100 ft by 100 ft?

    Example 10.66

    Calculating Distance with the Pythagorean Theorem

    The city has specific building codes for wheelchair ramps. Every vertical rise of 1 in requires that the horizontal length be 12 inches. You are constructing a ramp at your business. The plan is to make the ramp 130 inches in horizontal length and the slanted distance will measure approximately 132.4 inches (Figure 10.184). What should the vertical height be?

    A right triangle with its legs marked b and 130 inches. The hypotenuse is marked 132.4 inches.
    Figure 10.184
    Answer

    The Pythagorean Theorem states that the horizontal length of the base of the ramp, side a, is 130 in. The length of c, or the length of the hypotenuse, is 132.4 in. The length of the height of the triangle is side b.

    Then, by the Pythagorean Theorem, we have:

    a 2 + b 2 = c 2 ( 130 ) 2 + b 2 = ( 132.4 ) 2 16,900 + b 2 = 17,529.76 b 2 = 17,529.76 16,900 b 2 = 629.8 b = 629.8 = 25 a 2 + b 2 = c 2 ( 130 ) 2 + b 2 = ( 132.4 ) 2 16,900 + b 2 = 17,529.76 b 2 = 17,529.76 16,900 b 2 = 629.8 b = 629.8 = 25

    If you construct the ramp with a 25 in vertical rise, will it fulfill the building code? If not, what will have to change?

    The building code states 12 in of horizontal length for each 1 in of vertical rise. The vertical rise is 25 in, which means that the horizontal length has to be 12(25)=300in.12(25)=300in. So, no, this will not pass the code. If you must keep the vertical rise at 25 in, what will the other dimensions have to be? Since we need a minimum of 300 in for the horizontal length:

    ( 300 ) 2 + ( 25 ) 2 = c 2 90,625 = c 2 90,625 = c = 301 in ( 300 ) 2 + ( 25 ) 2 = c 2 90,625 = c 2 90,625 = c = 301 in

    The new ramp will look like Figure 10.185.

    A right triangle with its legs marked 25 inches and 300 inches. The hypotenuse is marked 301 inches.
    Figure 10.185

    Your Turn 10.66

    1.
    If 10 in is the maximum possible vertical rise as shown in the figure, how long would the ramp have to be to pass the building code rule of 12 horizontal inches to 1 vertical inch?
    A right triangle with its legs marked 10 inches and 120 inches. The hypotenuse is marked c.
    Figure 10.186

    30-60-9030-60-90 Triangles

    In geometry, as in all fields of mathematics, there are always special rules for special circumstances. An example is the perfect square rule in algebra. When expanding an expression like (2x+5y)2,(2x+5y)2, we do not have to expand it the long way:

    (2x+5y)2=(2x+5y)(2x+5y)=(2x)2+10xy+10xy+(5y)2=4x2+20xy+25y2(2x+5y)2=(2x+5y)(2x+5y)=(2x)2+10xy+10xy+(5y)2=4x2+20xy+25y2

    If we know the perfect square formula, given as

    (a+b)2=a2+2ab+b2,(a+b)2=a2+2ab+b2,

    we can skip the middle step and just start writing down the answer. This may seem trivial with problems like (a+b)2.Figure 10.187.

    A right triangle with its legs marked x and x times square root of 3. The hypotenuse is marked 2 x. The angles at the top, bottom-left, and bottom-right are labeled 60 degrees, 90 degrees, and 30 degrees.
    Figure 10.187 The 30-60-9030-60-90

    We see that the shortest side is opposite the smallest angle, and the longest side, the hypotenuse, will always be opposite the right angle. There is a set ratio of one side to another side for the 30-60-9030-60-90 triangle given as 1:3:2,1:3:2, or x:x3:2x.x:x3:2x. Thus, you only need to know the length of one side to find the other two sides in a 30-60-9030-60-90 triangle.

    Example 10.67

    Finding Missing Lengths in a 30-60-9030-60-90 Triangle

    Find the measures of the missing lengths of the triangle (Figure 10.188).

    A right triangle with its legs marked a and b. The hypotenuse is marked 10. The angles at the bottom-left and bottom-right are labeled 90 degrees and 30 degrees.
    Figure 10.188
    Answer

    We can see that this is a 30-60-9030-60-90 triangle because we have a right angle and a 3030 angle. The remaining angle, therefore, must equal 60.60. Because this is a special triangle, we have the ratios of the sides to help us identify the missing lengths. Side aa is the shortest side, as it is opposite the smallest angle 30,30, and we can substitute a=x.a=x. The ratios are x:x3:2x.x:x3:2x. We have the hypotenuse equaling 10, which corresponds to side cc, and side cc is equal to 2xx. Now, we must solve for xx:

    2 x = 10 x = 5 2 x = 10 x = 5

    Side bb is equal to x3x3 or 53.53. The lengths are 5,53,10.5,53,10.

    Your Turn 10.67

    1.
    Find the lengths of the missing sides in the given figure.
    A right triangle with its legs marked b and 15. The hypotenuse is marked c. The angles at the bottom-left and bottom-right are labeled 90 degrees and 60 degrees.
    Figure 10.189

    Example 10.68

    Applying 30-60-9030-60-90 Triangle to the Real World

    A city worker leans a 40-foot ladder up against a building at a 30Figure 10.190). How far up the building does the ladder reach?

    An illustration shows a 40-foot ladder placed against a building at an angle of 30 degrees. The vertical distance from the ladder to the ground is marked x. The horizontal distance from the base of the building to the base of the ladder is unknown.
    Figure 10.190
    Answer

    We have a 30-60-9030-60-90 triangle, and the hypotenuse is 40 ft. This length is equal to 2xx, where xx is the shortest side. If 2x=402x=40, then x=20x=20. The ladder is leaning on the wall 20 ft up from the ground.

    Your Turn 10.68

    1.
    You want to repair a window on the second floor of your home. If you place the ladder at a /**/{30^ \circ }/**/ angle to the ground, the ladder just about reaches the window. How far from the wall should you place the ladder? How far up will the ladder reach? Make a sketch as an aid.
    An illustration shows a 34-foot ladder placed against a building at an angle of 30 degrees. The vertical distance from the ladder to the ground is marked b. The horizontal distance from the base of the building to the base of the ladder is a.
    Figure 10.191

    45-45-9045-45-90 Triangles

    The 45-45-90Figure 10.192.

    Two right triangles. In the first triangle, the legs measure 1 and 1. The hypotenuse measures the square root of 2. The angles measure 90 degrees, 45 degrees, and 45 degrees. In the second triangle, the legs measure x and x. The hypotenuse measures x times the square root of 2. The angles measure 90 degrees, 45 degrees, and 45 degrees.
    Figure 10.192 45-45-9045-45-90 Triangles

    Example 10.69

    Finding Missing Lengths of a 45-45-9045-45-90 Triangle

    Find the measures of the unknown sides in the triangle (Figure 10.193).

    A right triangle. The legs measure a and 3. The hypotenuse measures c. The angles are marked 90 degrees, 45 degrees, and 45 degrees.
    Figure 10.193
    Answer

    Because we have a 45-45-9045-45-90 triangle, we know that the two legs are equal in length and the hypotenuse is a product of one of the legs and 2.2. One leg measures 3, so the other leg, aa, measures 3. Remember the ratio of x:x:x2.x:x:x2. Then, the hypotenuse, cc, equals 32.32.

    Your Turn 10.69

    1.
    Find the measures of the unknown sides in the given figure.
    A right triangle, A B C. The legs, C A, and A B measure x and x. The hypotenuse, C B measures 8. The angles, C and B are congruent. Angle A is a right angle.
    Figure 10.194

    Trigonometry Functions

    Trigonometry developed around 200 BC from a need to determine distances and to calculate the measures of angles in the fields of astronomy and surveying. Trigonometry is about the relationships (or ratios) of angle measurements to side lengths in primarily right triangles. However, trigonometry is useful in calculating missing side lengths and angles in other triangles and many applications.

    Checkpoint

    NOTE: You will need either a scientific calculator or a graphing calculator for this section. It must have the capability to calculate trigonometric functions and express angles in degrees.

    Trigonometry is based on three functions. We title these functions using the following abbreviations:

    • sin=sinesin=sine
    • cos=cosinecos=cosine
    • tan=tangenttan=tangent

    Letting r=x2+y2,Table 10.1. The functions are given in terms of xx, yy, and rr, and in terms of sides relative to the angle, like opposite, adjacent, and the hypotenuse.

    sinθ=yr=opphypsinθ=yr=opphyp cosθ=xr=adjhypcosθ=xr=adjhyp tanθ=yx=oppadjtanθ=yx=oppadj
    Table 10.1 Trigonometric Ratios

    We will be applying the sine function, cosine function, and tangent function to find side lengths and angle measurements for triangles we cannot solve using any of the techniques we have studied to this point. In Figure 10.195, we have an illustration mainly to identify rr and the sides labeled xx and yy.

    Two rays are plotted on an x y coordinate plane. Both rays begin at the origin. The first ray lies on the positive x-axis. The second ray lies in the first quadrant and a point, (x, y) is marked on the ray. The angle made by the two rays is marked theta. The distance from the origin to the point along the ray is labeled r.
    Figure 10.195 Angle θθ

    An angle θFigure 10.196, we will solve for the missing sides.

    Two rays are plotted on an x y coordinate plane. The ray lies in the first quadrant and a point is marked on the ray. A vertical line is drawn from the point to meet the x-axis and it measures y. The horizontal distance from the origin to the line is marked x. The angle made by the ray with the x-axis is marked 60 degrees. The distance from the origin to the point along the ray is labeled r equals 2.
    Figure 10.196 Solving for Missing Sides

    Let’s use the trigonometric functions to find the sides xx and yy. As long as your calculator mode is set to degrees, you do not have to enter the degree symbol. First, let’s solve for yy.

    We have sinθ=yr,sinθ=yr, and θ=60.θ=60. Then,

    sin60=y22sin60=y1.732=y3=ysin60=y22sin60=y1.732=y3=y

    Next, let’s find xx. This is the cosine function. We have cosθ=xr.cosθ=xr. Then,

    cos60=x22cos60=x=1cos60=x22cos60=x=1

    Now we have all sides, 1,3,2.Table 10.2 is a list of common angles, which you should find helpful.

    sin0=0sin0=0 cos0=1cos0=1
    sin30=12sin30=12 cos30=32cos30=32
    sin45=22sin45=22 cos45=22cos45=22
    sin60=32sin60=32 cos60=12cos60=12
    sin90=1sin90=1 cos90=0cos90=0
    Table 10.2 Common Angles

    Example 10.70

    Using Trigonometric Functions

    Find the lengths of the missing sides for the triangle (Figure 10.197).

    A ray is plotted on an x y coordinate plane. The ray lies in the first quadrant and a vertical dashed line is extended from a point on the ray to meet the x-axis and it makes a right angle. The horizontal distance from the origin to the line is marked 6. The angle made by the ray with the x-axis is marked 55 degrees. The distance from the origin to the point along the ray is labeled r.
    Figure 10.197
    Answer

    We have a 5555 angle, and the length of the triangle on the xx-axis is 6 units.

    Step 1: To find the length of rr, we can use the cosine function, as cosθ=xr.cosθ=xr. We manipulate this equation a bit to solve for rr:

    cos ( 55 ) = 6 r r cos ( 55 ) = 6 r = 6 cos ( 55 ) r = 6 0.5736 = 10.46 cos ( 55 ) = 6 r r cos ( 55 ) = 6 r = 6 cos ( 55 ) r = 6 0.5736 = 10.46

    Step 2: We can use the Pythagorean Theorem to find the length of yy. Prove that your answers are correct by using other trigonometric ratios:

    6 2 + y 2 = 10.46 2 y 2 = 109.4 36 y = 8.57 6 2 + y 2 = 10.46 2 y 2 = 109.4 36 y = 8.57

    Step 3: Now that we have yy, we can use the sine function to prove that rr is correct. We have sinθ=yr.sinθ=yr.

    sin ( 55 ) = 8.57 r r sin ( 55 ) = 8.57 r = 8.57 sin ( 55 ) = 8.57 0.819 = 10.46 sin ( 55 ) = 8.57 r r sin ( 55 ) = 8.57 r = 8.57 sin ( 55 ) = 8.57 0.819 = 10.46

    Your Turn 10.70

    1.
    Find the lengths of the missing sides in the given figure.
    A ray is plotted on an x y coordinate plane. The ray lies in the first quadrant and a vertical dashed line is extended from a point on the ray to meet the x-axis and it makes a right angle. The horizontal distance from the origin to the line is marked 5. The angle made by the ray with the x-axis is marked 40 degrees. The distance from the origin to the point along the ray is labeled r.
    Figure 10.198

    To find angle measurements when we have two side measurements, we use the inverse trigonometric functions symbolized as sin1,sin1, cos1,cos1, or tan1.tan1. The –1 looks like an exponent, but it means inverse. For example, in the previous example, we had x=6x=6 and r=10.46.r=10.46. To find what angle has these values, enter the values for the inverse cosine function cos1(xr)cos1(xr) in your calculator:

    cos1(610.46)=55.cos1(610.46)=55.

    You can also use the inverse sine function and enter the values of sin1(yr)sin1(yr) in your calculator given y=8.57y=8.57 and r=10.46.r=10.46. We have

    sin1(8.5710.46)=55.sin1(8.5710.46)=55.

    Finally, we can also use the inverse tangent function. Recall tanθ=yx.tanθ=yx. We have

    tan1(8.576)=55.tan1(8.576)=55.

    Example 10.71

    Solving for Lengths in a Right Triangle

    Solve for the lengths of a right triangle in which θ=30Figure 10.199).

    A ray is plotted on an x y coordinate plane. The ray lies in the first quadrant and a vertical dashed line is extended from a point on the ray to meet the x-axis and it makes a right angle. This dashed line measures a. The horizontal distance from the origin to the line is marked b. The angle made by the ray with the x-axis is marked 30 degrees. The distance from the origin to the point along the ray is labeled 6.
    Figure 10.199
    Answer

    Step 1: To find side aa, we use the sine function:

    sin 30 = a 6 6 sin 30 = a = 3 sin 30 = a 6 6 sin 30 = a = 3

    Step 2: To find bb, we use the cosine function:

    cos 30 = b 6 6 cos 30 = b = 5.196 cos 30 = b 6 6 cos 30 = b = 5.196

    Step 3: Since this is a 30-60-9030-60-90 triangle and side bb should equal x3,x3, if we input 3 for xx, we have b=33.b=33. Put this in your calculator and you will get 33=5.196.33=5.196.

    Your Turn 10.71

    1.
    Find the missing side and angles in the figure shown.
    A ray is plotted on an x y coordinate plane. The ray lies in the first quadrant and a vertical dashed line is extended from a point on the ray to meet the x-axis and it makes a right angle. This dashed line measures 6. The horizontal distance from the origin to the line is marked x. The angle made by the ray with the x-axis is marked alpha. The distance from the origin to the point along the ray is labeled 8.3. The angle made by the ray and the dashed line is marked beta.
    Figure 10.200

    Example 10.72

    Finding Altitude

    A small plane takes off from an airport at an angle of 31.3Figure 10.201). If the plane continues that angle of ascent, find its altitude when it is above the peak, and how far it will be above the peak.

    A horizontal line measures 3520 feet. A ray originates from the left end of the line and it makes an angle of 31.3 degrees. A vertical line measuring 1100 feet is on the right end of the line. A plane is flying above the line.
    Figure 10.201
    Answer

    To solve this problem, we use the tangent function:

    tan 31.3 = x 3,520 3,520 tan 31.3 = 2,140 tan 31.3 = x 3,520 3,520 tan 31.3 = 2,140

    The plane’s altitude when passing over the peak is 2,140 ft, and it is 1,040 ft above the peak.

    Your Turn 10.72

    1.
    Suppose that the plane takes off at a /**/{23^ \circ }/**/ angle, and 1 mile from the airport is a 1,500-foot peak. At what altitude will the plane pass over the peak?

    Example 10.73

    Finding Unknown Sides and Angles

    Suppose you have two known sides, but do not know the measure of any angles except for the right angle (Figure 10.202). Find the measure of the unknown angles and the third side.

    A right triangle with its legs marked 4 and 6. The hypotenuse is marked c. The angle made by the hypotenuse and the horizontal leg is marked theta.
    Figure 10.202
    Answer

    Step 1: We can find the third side using the Pythagorean Theorem:

    6 2 + 4 2 = c 2 52 = c 2 2 13 = c 6 2 + 4 2 = c 2 52 = c 2 2 13 = c

    Now, we have all three sides.

    Step 2: To find θ,θ, we will first find sinθ.sinθ.

    sin θ = o p p h y p = 4 2 13 = 2 13 . sin θ = o p p h y p = 4 2 13 = 2 13 .

    The angle θθ is the angle whose sine is 213.213.

    Step 3: To find θθ, we use the inverse sine function:

    θ = sin 1 ( 2 13 ) = 33.7 θ = sin 1 ( 2 13 ) = 33.7

    Step 4: To find the last angle, we just subtract: 1809033.7=56.31809033.7=56.3.

    Your Turn 10.73

    1.
    You know the lengths of two sides and the right angle as shown in the figure. Find the length of the third side and the other angles.
    A right triangle with its legs marked 7 and 4. The hypotenuse is marked c. The angle made by the hypotenuse and the horizontal leg is marked.
    Figure 10.203

    Angle of Elevation and Angle of Depression

    Other problems that involve trigonometric functions include calculating the angle of elevation and the angle of depression. These are very common applications in everyday life. The angle of elevation is the angle formed by a horizontal line and the line of sight from an observer to some object at a higher level. The angle of depression is the angle formed by a horizontal line and the line of sight from an observer to an object at a lower level.

    Example 10.74

    Finding the Angle of Elevation

    A guy wire of length 110 meters runs from the top of an antenna to the ground (Figure 10.204). If the angle of elevation of an observer to the top of the antenna is 43,43, how high is the antenna?

    An illustration shows a right triangle. The vertical leg of the triangle represents the height of the antenna. The hypotenuse represents a guy wire of 110 meters. An observer is at the bottom-left vertex of the triangle. The bottom-left and bottom-right triangles are marked 43 degrees and 90 degrees.
    Figure 10.204
    Answer

    We are looking for the height of the tower. This corresponds to the yy-value, so we will use the sine function:

    sin 43 = y 110 110 sin 43 = y 75 = y sin 43 = y 110 110 sin 43 = y 75 = y

    The tower is 75 m high.

    Your Turn 10.74

    1.
    You travel to Chicago and visit the observation deck at Willis Tower, 1,450 ft above ground. You can see the Magnificent Mile to the northeast 6,864 ft away. What is the angle of depression from the observation deck to the Magnificent Mile?
    An illustration shows a right triangle. The vertical leg of the triangle represents the height of the Willis Tower and it measures 1450 feet. The horizontal leg measures 6864 feet. A horizontal dashed line is drawn from the top of the triangle. The angle between the hypotenuse and the horizontal dashed line represents the angle of depression.
    Figure 10.205

    Example 10.75

    Finding Angle of Elevation

    You are sitting on the grass flying a kite on a 50-foot string (Figure 10.206). The angle of elevation is 60.60. How high above the ground is the kite?

    A right triangle. A kite is placed at the top of the triangle. The hypotenuse measures 50 feet. The angle made by the hypotenuse and the horizontal leg measures 60 degrees.
    Figure 10.206
    Answer

    We can solve this using the sine function, sinθ=opphyp.sinθ=opphyp.

    sin 60 = x 50 50 sin 60 = x = 43.3 ft sin 60 = x 50 50 sin 60 = x = 43.3 ft

    Your Turn 10.75

    1.
    You are flying a kite on a 60-foot string. The angle of elevation from the ground to the kite is /**/{50^ \circ}/**/. How high above the ground is the kite?
    A right triangle. A kite is placed at the top of the triangle. The hypotenuse measures 60 feet. The angle made by the hypotenuse and the horizontal leg measures 50 degrees.
    Figure 10.207

    People in Mathematics

    Pythagoras and the Pythagoreans

    The Pythagorean Theorem is so widely used that most people assume that Pythagoras (570–490 BC) discovered it. The philosopher and mathematician uncovered evidence of the right triangle concepts in the teachings of the Babylonians dating around 1900 BC. However, it was Pythagoras who found countless applications of the theorem leading to advances in geometry, architecture, astronomy, and engineering.

    Among his accolades, Pythagoras founded a school for the study of mathematics and music. Students were called the Pythagoreans, and the school’s teachings could be classified as a religious indoctrination just as much as an academic experience. Pythagoras believed that spirituality and science coexist, that the intellectual mind is superior to the senses, and that intuition should be honored over observation.

    Pythagoras was convinced that the universe could be defined by numbers, and that the natural world was based on mathematics. His primary belief was All is Number. He even attributed certain qualities to certain numbers, such as the number 8 represented justice and the number 7 represented wisdom. There was a quasi-mythology that surrounded Pythagoras. His followers thought that he was more of a spiritual being, a sort of mystic that was all-knowing and could travel through time and space. Some believed that Pythagoras had mystical powers, although these beliefs were never substantiated.

    Pythagoras and his followers contributed more ideas to the field of mathematics, music, and astronomy besides the Pythagorean Theorem. The Pythagoreans are credited with the discovery of irrational numbers and of proving that the morning star was the planet Venus and not a star at all. They are also credited with the discovery of the musical scale and that different strings made different sounds based on their length. Some other concepts attributed to the Pythagoreans include the properties relating to triangles other than the right triangle, one of which is that the sum of the interior angles of a triangle equals 180.180. These geometric principles, proposed by the Pythagoreans, were proven 200 years later by Euclid.

    Who Knew?

    A Visualization of the Pythagorean Theorem

    In Figure 10.208, which is one of the more popular visualizations of the Pythagorean Theorem, we see that square aa is attached to side aa; square bb is attached to side bb; and the largest square, square cc, is attached to side cc. Side aa measures 3 cm in length, side bb measures 4 cm in length, and side cc measures 5 cm in length. By definition, the area of square aa measures 9 square units, the area of square bb measures 16 square units, and the area of square cc measures 25 square units. Substitute the values given for the areas of the three squares into the Pythagorean Theorem and we have

    a2+b2=c232+42=529+16=25a2+b2=c232+42=529+16=25

    Thus, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, as stated in the Pythagorean Theorem.

    A right triangle with its legs marked a equals 3 and b equals 4. The hypotenuse is marked c equals 5. A square is placed along each side.
    Figure 10.208

    Check Your Understanding

    48.
    Find the lengths of the unknown sides of the /**/{30^ \circ } {\text{-}} {60^ \circ } {\text{-}} {90^ \circ }/**/ triangle shown..
    A right triangle. The legs are labeled a and b. The hypotenuse is labeled 5. The angles at the bottom-left and bottom-right are labeled 30 degrees and 90 degrees.
    49.
    Find the missing lengths of the /**/{45^ \circ } {\text{-}} {45^ \circ } {\text{-}} {90^ \circ }/**/ triangle shown.
    A right triangle. The legs are labeled 10 and b. The hypotenuse is labeled c. The angles at the top, bottom-left, and bottom-right are labeled 45 degrees, 90 degrees, and 45 degrees.
    50.
    Use the Pythagorean theorem to find the missing length in the triangle shown.
    A right triangle. The legs are labeled b and 3. The hypotenuse is labeled 3 times square root of 5.
    51.
    The sun casts a shadow over the roof of a house that ends 105 ft from the front door as shown in the figure. How high is the house to the tip of the roof?
    An illustration shows a right triangle. The vertical leg represents the height of a house. The sun is above the house. The horizontal leg measures 105 feet. The angle formed by the hypotenuse and the horizontal leg measures 13 degrees.
    52.
    Find the measure of side /**/c/**/ in the given figure.
    A right triangle. The legs are labeled 7.5 centimeters and unknown. The hypotenuse is labeled c. The angles at the bottom-left and bottom-right are 90 degrees and 40 degrees.
    53.
    Find the measure of side /**/x/**/ in the given figure.
    A right triangle. The legs are labeled unknown and x. The hypotenuse is labeled 15 meters. The angles at the bottom-left and bottom-right are 90 degrees and 18 degrees.

    Section 10.8 Exercises

    Use the Pythagorean theorem to answer the following exercises. Let /**/a/**/ and /**/b/**/ represent two legs of a right triangle, and let /**/c/**/ represent the hypotenuse. Find the lengths of the missing sides.
    1.
    A right triangle. The legs are labeled a equals 7 and b. The hypotenuse is labeled c equals 25.
    2.
    A right triangle. The legs are labeled a equals 8 and b equals 15. The hypotenuse is labeled c.
    3.
    If /**/b = 25/**/ and /**/c = 65{\text{ ft}}/**/, find /**/a/**/.
    4.
    If /**/a = 18{\text{ ft}}/**/ and /**/c = 30{\text{ ft}}/**/, find /**/b/**/.
    5.
    If /**/b = 12{\text{ cm}}/**/ and /**/c = 15{\text{ cm}}/**/, find /**/a/**/.
    Refer to the /**/{30^ \circ} {\text{-}} {60^ \circ } {\text{-}} {90^ \circ }/**/ triangle shown for the following exercises.
    A right triangle. The legs are labeled a and b. The hypotenuse is labeled c. The angles at the top, bottom-left, and bottom-right are labeled 60 degrees, 90 degrees, and 30 degrees.
    6.
    If /**/b = 10/**/, find /**/c/**/ and /**/a/**/
    7.
    If /**/a = 2\sqrt 3{\text{ ft}}/**/, find /**/b/**/ and /**/c/**/.
    8.
    If /**/b = 7{\text{ cm}}/**/, find /**/a/**/ and /**/c/**/
    9.
    If /**/c = \sqrt 3 /**/, find /**/a/**/ and /**/b/**/.
    10.
    If /**/c = 8\sqrt 3 /**/, find /**/a/**/ and /**/b/**/
    Refer to the /**/{45^ \circ }{\text{-}}{45^ \circ }{\text{-}}{90^ \circ }/**/ triangle shown in the following exercises.
    A right triangle. The legs are labeled a and b. The hypotenuse is labeled c. The angles at the top, bottom-left, and bottom-right are labeled 90 degrees, 45 degrees, and 45 degrees.
    11.
    If /**/a = 15{\text{ cm}}/**/, find /**/b/**/.
    12.
    If /**/a = 3\sqrt {2}{\text{ ft}}/**/, find /**/a/**/.
    13.
    If /**/a = 4\sqrt 5{\text{ cm}}/**/, find /**/b/**/.
    14.
    If /**/a = 10{\text{ ft}}/**/, find /**/b/**/.
    15.
    If /**/b = \frac
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    , find /**/a/**/.
    16.
    In the provided figure given /**/\alpha = {10^ \circ }/**/, find the unknown sides and angles.
    A right triangle. The legs are labeled a equals 11 and b. The angles at the top, bottom-left, and bottom-right are labeled beta, 90 degrees, and alpha.
    Evaluate the expressions in the following exercises.
    17.
    /**/{\sin ^{ - 1}}\left( {\frac{2}{5}} \right)/**/
    18.
    /**/{\tan ^{ - 1}}(5.68)/**/
    19.
    /**/{\cos ^{ - 1}}\left( {\frac{1}{9}} \right)/**/
    20.
    Use the figure shown to solve for all missing sides and angles given /**/a = 6,\,b = 8/**/.
    A right triangle. The legs are labeled b and a. The hypotenuse is labeled c. The angles at the top, bottom-left, and bottom-right are labeled beta, alpha, and 90 degrees.
    21.
    Use the figure shown to solve for all missing sides and angles given /**/\alpha = {16^ \circ },\,c = 20/**/.
    A right triangle. The legs are labeled b and a. The hypotenuse is labeled 20. The angles at the top, bottom-left, and bottom-right are labeled beta, 90 degrees, and 16 degrees.

    This page titled 10.9: Right Triangle Trigonometry is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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