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1.E: Right Triangle Trigonometry Angles (Exercises)

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These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.

1.1 Exercises

For Exercises 1-4, find the numeric value of the indicated angle(s) for the triangle .

1.1.1 Find if and .

1.1.2 Find if and .

1.1.3 Find and if , , and .

1.1.4 Find , , and if and .

For Exercises 5-8, find the numeric value of the indicated angle(s) for the right triangle , with being the right angle.

1.1.5 Find if .

1.1.6 Find and if and .

1.1.7 Find and if and .

1.1.8 Find and if and .

1.1.9 A car goes 24 miles due north then 7 miles due east. What is the straight distance between the car's starting point and end point?

1.1.10 One end of a rope is attached to the top of a pole 100 ft high. If the rope is 150 ft long, what is the maximum distance along the ground from the base of the pole to where the other end can be attached? You may assume that the pole is perpendicular to the ground.

1.1.11 Prove that the hypotenuse is the longest side in every right triangle. (Hint: Is ?)

1.1.12 Can a right triangle have sides with lengths 2, 5, and 6? Explain your answer.

1.1.13 If the lengths , , and of the sides of a right triangle are positive integers, with , then they form what is called a Pythagorean triple. The triple is normally written as (,,). For example, (3,4,5) and (5,12,13) are well-known Pythagorean triples.
(a) Show that (6,8,10) is a Pythagorean triple.
(b) Show that if (,,) is a Pythagorean triple then so is (,,) for any integer . How would you interpret this geometrically?
(c) Show that (,,) is a Pythagorean triple for all integers .
(d) The triple in part(c) is known as Euclid's formula for generating Pythagorean triples. Write down the first ten Pythagorean triples generated by this formula, i.e. use: and ; and , ; and , , ; and , , , .

1.1.14 This exercise will describe how to draw a line through any point outside a circle such that the line intersects the circle at only one point. This is called a tangent line to the circle (see the picture on the left in Figure 1.1.6), a notion which we will use throughout the text.

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Figure 1.1.6

On a sheet of paper draw a circle of radius 1 inch, and call the center of that circle . Pick a point which is inches away from . Draw the circle which has as a diameter, as in the picture on the right in Figure 1.1.6. Let be one of the points where this circle intersects the first circle. Draw the line through and . In general the tangent line through a point on a circle is perpendicular to the line joining that point to the center of the circle (why?). Use this fact to explain why the line you drew is the tangent line through and to calculate the length of . Does it match the physical measurement of ?

1.1.15 Suppose that is a triangle with side a diameter of a circle with center , as in the picture on the right, and suppose that the vertex lies on the circle. Now imagine that you rotate the circle around its center, so that is in a new position, as indicated by the dashed lines in the picture. Explain how this picture proves Thales' Theorem.

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1.2 Exercises

For Exercises 1-10, find the values of all six trigonometric functions of\\angles and in the right triangle in Figure 1.2.3.

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Figure 1.2.3

1.2.1 , ,

1.2.2 , ,

1.2.3 , ,

1.2.4 , ,

1.2.5 , ,

1.2.6 , ,

1.2.7 ,

1.2.8 ,

1.2.9 ,

1.2.10 ,

For Exercises 11-18, find the values of the other five trigonometric functions of the acute angle given the indicated value of one of the functions.

1.2.11

1.2.12

1.2.13

1.2.14

1.2.15

1.2.16

1.2.17

1.2.18

For Exercises 19-23, write the given number as a trigonometric function of an acute angle less than .

1.2.19

1.2.20

1.2.21

1.2.22

1.2.23

For Exercises 24-28, write the given number as a trigonometric function of an acute angle greater than .

1.2.24

1.2.25

1.2.26

1.2.27

1.2.28

1.2.29 In Example 1.7 we found the values of all six trigonometric functions of and .

(a) Does ?
(b) Does ?
(c) Does ?
(d) Does ?

1.2.30 For an acute angle , can be larger than ? Explain your answer.

1.2.31 For an acute angle , can be larger than ? Explain your answer.

1.2.32 For an acute angle , can be larger than ? Explain your answer.

1.2.33 If and are acute angles and , explain why .

1.2.34 If and are acute angles and , explain why .

1.2.35 Prove the Cofunction Theorem (Theorem 1.2). (Hint: Draw a right triangle and label the angles and sides.)

1.2.36 Use Example 1.10 to find all six trigonometric functions of .

1.2.37 In Figure 1.2.4, is a diameter of a circle with a radius of cm and center , is a right triangle, and has length cm.

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Figure 1.2.4

(a) Find . (Hint: Use Thales' Theorem.)
(b) Find the length of .
(c) Find the length of .
(d) Figure 1.2.4 is drawn to scale. Use a protractor to measure the angle , then use your calculator to find\\the sine of that angle. Is the calculator result close to your answer from part(a)? Note: Make sure that your calculator is in degree mode.


1.2.38 In Exercise 37, verify that the area of equals . Why does this make sense?

1.2.39 In Exercise 37, verify that the area of equals .

1.2.40 In Exercise 37, verify that the area of equals .


1.3 Exercises

1.3.1 From a position ft above the ground, an observer in a building measures angles of depression of and to the top and bottom, respectively, of a smaller building, as in the picture on the right. Use this to find the height of the smaller building.


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1.3.2 Generalize Example 1.12: A person standing ft from the base of a mountain measures an angle of elevation from the ground to the top of the mountain. The person then walks ft straight back and measures an angle of elevation to the top of the mountain, as in the picture on the right. Assuming the ground is level, find a formula for the height of the mountain in terms of , , , and .

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1.3.3 As the angle of elevation from the top of a tower to the sun decreases from to during the day, the length of the shadow of the tower increases by ft along the ground. Assuming the ground is level, find the height of the tower.

1.3.4 Two banks of a river are parallel, and the distance between two points and along one bank is ft. For a point on the opposite bank, and , as in the picture on the right. What is the width of the river? (Hint: Divide into two pieces.)

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1.3.5 A tower on one side of a river is directly east and north of points and , respectively, on the other side of the river. The top of the tower has angles of elevation and from and , respectively, as in the picture on the right. Let be the distance between and . Assuming that both sides of the river are at the same elevation, show that the height of the tower is

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1.3.6 The equatorial parallax of the moon has been observed to be approximately . Taking the radius of the earth to be miles, estimate the distance from the center of
the earth to the moon. (Hint: See Example 1.15.)

1.3.7 An observer on earth measures an angle of from one visible edge of the moon to the other (opposite) edge. Use this to estimate the radius of the moon. (Hint: Use Exercise
6 and see Example 1.16.
)

1.3.8 A ball bearing sits between two metal grooves, with the top groove having an angle of and the bottom groove having an angle of , as in the picture on the right. What must the diameter of the ball bearing be for the distance between the vertexes of the grooves to be half an inch? You may assume that the top vertex is directly above the bottom vertex.

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1.3.9 The machine tool diagram on the right shows a symmetric worm thread, in which a circular roller of diameter inches sits. Find the amount that the top of the roller rises above the top of the thread, given the information in the diagram. (Hint: Extend the slanted sides of the thread until they meet at a point.)

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1.3.10 Repeat Exercise 9 using inches as the distance across the top of the worm thread.

1.3.11 In Exercise 9, what would the distance across the top of the worm thread have to be to make equal to inches?

1.3.12 For in the slider-crank mechanism in Example 1.18, show that


(Hint: In Figure 1.3.2 draw line segments from perpendicular to and .)

1.3.13 The machine tool diagram on the right shows a symmetric die punch. In this view, the rounded tip is part of a circle of radius , and the slanted sides are tangent to that circle and form an angle of . The top and bottom sides of the die punch are horizontal. Use the
information in the diagram to find the radius .

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1.3.14 In the figure on the right, and . Use this to find , , , , , and in terms of and .(Hint: What is the angle ?)

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For Exercises 15-23, solve the right triangle in Figure 1.3.4 using the given information.

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Figure 1.3.4

1.3.15 ,

1.3.16 ,

1.3.17 ,

1.3.18 ,

1.3.19 ,

1.3.20 ,

1.3.21 ,

1.3.22 ,

1.3.23 ,

1.3.24 In Example 1.10 in Section 1.2, we found the exact values of all six trigonometric functions of . For example, we showed that . So since by the Cofunction Theorem, this means that . We will now describe another method for finding the exact values of the trigonometric functions of . In fact, it can be used to find the exact values for the trigonometric functions of when those for are known, for any . The method is illustrated in Figure 1.3.5 and is described below.

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Figure 1.3.5

Draw a semicircle of radius centered at a point on a horizontal line. Let be the point on the semicircle such that makes an angle of with the horizontal line, as in Figure 1.3.5. Draw a line straight down from to the horizontal line at the point . Now create a second semicircle as follows: Let be the left endpoint of the first semicircle, then draw a new semicircle centered at with radius equal to . Then create a third semicircle in the same way: Let be the left endpoint of the second semicircle, then draw a new semicircle centered at with radius equal to .

This procedure can be continued indefinitely to create more semicircles. In general, it can be shown that the line segment from the center of the new semicircle to makes an angle with the horizontal line equal to half the angle from the previous semicircle's center to .

(a) Explain why . (Hint: What is the supplement of ?)
(b) Explain why and .
(c) Use Figure 1.3.5 to find the exact values of , , and . (Hint: To start, you will need to use and to find the exact lengths of and .)
(d) Use Figure 1.3.5 to calculate the exact value of .
(e) Use the same method but with an initial angle of to find the exact values of , , and .

1.3.25 A manufacturer needs to place ten identical ball bearings against the inner side of a circular container such that each ball bearing touches two other ball bearings, as in the picture on the right. The (inner) radius of the container is cm.

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(a) Find the common radius of the ball bearings.
(b) The manufacturer needs to place a circular ring\\inside the container. What is the largest possible (outer) radius of the ring such that it is not on top\\of the ball bearings and its base is level with the\\base of the container?

1.3.26 A circle of radius is inscribed inside a polygon with eight sides of equal length, called a regular octagon. That is, each of the eight sides is tangent to the circle, as in the picture on the right.

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(a) Calculate the area of the octagon.
(b) If you were to increase the number of sides of the\\polygon, would the area inside it increase or decrease? What number would the area approach, if any? Explain.
(c) Inscribe a regular octagon inside the same circle. That is, draw a regular octagon such that each of its eight vertexes touches the circle. Calculate the area of this octagon.

1.3.27 The picture on the right shows a cube whose sides are of length .

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(a) Find the length of the diagonal line segment .
(b) Find the angle that makes with the base of the cube.

1.3.28 In Figure 1.3.6, suppose that , , and are known. Show that:

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Figure 1.3.6

(a)
(b)
(c)
(Hint: What is the measure of the angle ?)

1.3.29 Persons A and B are at the beach, their eyes are ft and ft, respectively, above sea level. How many miles farther out is Person B's horizon than Person A's? (Note: mile = ft)


1.4 Exercises

For Exercises 1-10, state in which quadrant or on which axis the given angle lies.

1.4.1

1.4.2

1.4.3

1.4.4

1.4.5

1.4.6

1.4.7

1.4.8

1.4.9

1.4.10

1.4.11 In which quadrant(s) do sine and cosine have the same sign?

1.4.12 In which quadrant(s) do sine and cosine have the opposite sign?

1.4.13 In which quadrant(s) do sine and tangent have the same sign?

1.4.14 In which quadrant(s) do sine and tangent have the opposite sign?

1.4.15 In which quadrant(s) do cosine and tangent have the same sign?

1.4.16 In which quadrant(s) do cosine and tangent have the opposite sign?

For Exercises 17-21, find the reference angle for the given angle.

1.4.17

1.4.18

1.4.19

1.4.20

1.4.21

For Exercises 22-26, find the exact values of and when has the indicated value.

1.4.22

1.4.23

1.4.24

1.4.25

1.4.26

For Exercises 27-31, find the exact values of and when has the indicated value.

1.4.27

1.4.28

1.4.29

1.4.30

1.4.31

For Exercises 32-36, find the exact values of and when has the indicated value.

1.4.32

1.4.33

1.4.34

1.4.35

1.4.36

For Exercises 37-40, use Table 1.3 to answer the following questions.

1.4.37 Does ?

1.4.38 Does ?

1.4.39 Does ?

1.4.40 Does ?

1.4.41 Expand Table 1.3 to include all integer multiples of . See Example 1.10 in Section 1.2.


1.5 Exercises

1.5.1 Let . Find the angle between and which is the

(a) reflection of around the -axis
(b) reflection of around the -axis
(c) reflection of around the origin

1.5.2 Repeat Exercise 1 with .

1.5.3 Repeat Exercise 1 with .

1.5.4 We proved Equations 1.4-1.6 for any angle in QI. Mimic that proof to show that the formulas hold for in QII.

1.5.5 Verify Equations 1.4-1.6 for on the coordinate axes, i.e. for , , , .

1.5.6 In Example 1.26 we used the formulas involving to prove that the slopes of perpendicular lines are negative reciprocals. Show that this result can also be proved using the formulas involving . (Hint: Only the last paragraph in that example needs to be modified.)

For Exercises 7-14, find all angles which satisfy the given equation:

1.5.7

1.5.8

1.5.9

1.5.10

1.5.11

1.5.12

1.5.13

1.5.14

1.5.15 In our proof of the Pythagorean Theorem in Section 1.2, we claimed that in a right triangle it was possible to draw a line segment from the right angle vertex to a point on the hypotenuse such that . Use the picture below to prove that claim. (Hint: Notice how is placed on the -coordinate plane. What is the slope of the hypotenuse? What would be the slope of a line perpendicular to it?) Also, find the coordinates of the point in terms of and .

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1.5.16 It can be proved without using trigonometric functions that the slopes of perpendicular lines are negative reciprocals. Let and be perpendicular lines (with nonzero slopes), as in the picture below. Use the picture to show that .(Hint: Think of similar triangles and the definition of slope.)

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1.5.17 Prove Equations 1.19-1.21 by using Equations 1.10-1.12 and 1.13-1.15.


This page titled 1.E: Right Triangle Trigonometry Angles (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by Michael Corral via source content that was edited to the style and standards of the LibreTexts platform.

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