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Mathematics LibreTexts

5.2: Right Triangle Trigonometry

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    34917
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    Right Triangle Relationships

    In this section we introduce right triangle trigonometry and explore the applications of the trigonometric functions to right triangles where circles may not be involved. In later sections we will examine trigonometric functions as ratios of the sides of a triangle drawn inside a circle, and discuss the role of those functions in finding points on the circle. 

    Trigonometric functions are ratios of two of the three sides of a right triangle. The longest side of the right triangle is the hypotenuse. The other two sides are defined in terms of their relationship to one of the acute angles in the right triangle -- the side opposite the chosen angle, and the side adjacent (next to) that angle. The triangle can be oriented any where in space, but when the triangle is placed in standard position with the chosen acute angle \( \theta\) at the origin, these sides can be expressed equivalently in terms of \(x\) for the opposite side, \(y\) for the adjacent side, and \(r\) for the hypotenuse. The point \( (x, y) \) is a point on the terminal side of the angle.

    RIGHT TRIANGLE RELATIONSHIPS

    Given a right triangle with an angle of \(\theta\)

    \( \begin{array}{rllrll} 
    \sin \theta &= \dfrac{\text{opposite}}{\text{hypotenuse}} &= \dfrac{y}{r}  \qquad &\csc \theta &= \dfrac{\text{hypotenuse}}{\text{opposite}} &= \dfrac{r}{y}  \\
    \cos \theta &= \dfrac{\text{adjacent}}{\text{hypotenuse}} &= \dfrac{x}{r}  \qquad &\sec \theta &= \dfrac{\text{hypotenuse}}{\text{adjacent}} &= \dfrac{r}{x}  \\
    \tan \theta &= \dfrac{\text{opposite}}{\text{adjacent}} &= \dfrac{y}{x}  \qquad &\cot \theta &= \dfrac{\text{adjacent}}{\text{opposite}}  &= \dfrac{x}{y} \\
    \end{array}   \)

    A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse, Cosine is adjacent over hypotenuse, Tangent is opposite over adjacent.”

    Example \(\PageIndex{1}\): Find trigonometric ratios given 3 sides of a right triangle

    Given the triangle shown, find the value for \(\text{cos}(\alpha)\).屏幕快照 2019-07-08 上午6.53.30.png

    The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so

    \[\text{cos}(\alpha) = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{15}{17}\nonumber\]

     Try It \(\PageIndex{1}\)

    A right triangle with sides of 7, 24, and 25. Also labeled is angle t which is opposite the side labeled 7.Given the triangle shown, find the value of \(\sin t\).

     
    Answer
    \(\sin t = \dfrac{7}{25}\)

    It is not necessary that the lengths of all three sides of the right triangle be known initially. The Theorem of Pythagoras can be used to find the third side if we are given the lengths of only two sides of a right triangle.

    Example \(\PageIndex{2}\): Find trigonometric ratios given 2 sides of a right triangle

    Given a triangle with a hypotenuse of \(9\) and side opposite to angle \( \alpha\) of \(5\), find the sine, cosine and tangent for angle \( \alpha\).

    5.2 example triangle.pngSolution.

    The triangle with the given information is illustrated on the right. The third side, which in this case is the "adjacent" side, can be found by using the Theorem of Pythagoras \( a^2 + b^2 = c^2\). Always remember that in the formula, \(c\) is the length of the hypotenuse. From \( x^2 + 5^2 = 9^2\) we obtain \(x^2 = 81-25 = 56\). So \(x = \sqrt{56} = 2\sqrt{14}\). 

    Now that all three sides have been found, values for the trigonometric functions can be calculated.

    \( \sin \alpha = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{2\sqrt{14}}{9} \) \( \qquad\)  \( \cos \alpha = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{5}{9} \) \( \qquad\)  \( \tan \alpha = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{2\sqrt{14}}{5} \)

     Try It \(\PageIndex{2}\)

    Given a triangle with the side adjacent to angle \(\theta\) being \(x=2\) and the side opposite angle \(\theta\) being \(x=5\), find the secant, cosecant, and cotangent for angle \(theta\).

    Answer

    From \(x^2 + y^2 = r^2\) we obtain \( r = \sqrt{x^2+y^2}\) and so \(r = \sqrt{29}\).

    Therefore, \( \csc \theta = \dfrac{\sqrt{29}}{5} ,   \; \sec \theta = \dfrac{\sqrt{29}}{2},   \; \cot \theta = \dfrac{5}{2},  \)

    5.2similar triangle diagram.pngGiven One Trigonometric Ratio, find all others

    One important point to make here is that it is only the angle that determines the values of these trigonometric ratios, not the size of the triangle being used to determine these ratios. This is because triangles constructed with the same angles but different length sides  are similar triangles, so their ratios evaluated will be the same.

    Thus, \(  \dfrac{y_{1} }{r_{1} } =\dfrac{y_{2} }{r_{2} } \qquad \dfrac{x_{1} }{r_{1} } =\dfrac{x_{2} }{r_{2} }\)

    A trigonometric ratio is the ratio of two sides of a right triangle. Since right triangles that are similar can be used to establish values for trigonometric functions,  the numerator of the trigonometric ratio can serve as the length of one of the sides; the denominator can serve as the length of another side. Whether these sides are the hypotenuse of the triangle or opposite or adjacent to the given angle depends on which trigonometric ratio is being used.

    Example \(\PageIndex{3}\): Find all trigonometric ratios given one ratio

    Given \( \sec \theta = 4 \), find the other five trigonometric ratios for angle \( \theta \).

    Solution

    G5.4text triangle example  csc is 4.pngiven \( \sec \theta = 4 \), we can construct the ratio \( \sec \theta = \dfrac{4}{1} =  \dfrac{\text{hypotenuse}}{\text{adjacent}} = \dfrac{r}{x} \).

    Drawing a triangle can help to visualize the relationships represented by this trigonometric ratio.

    Using the Theorem of Pythagoras \( x^2 +y^2 = r^2\) and knowing \(r = 4\), \(x = 1\), we obtain \( 1^2 + y^2 = 4^2\).  Thus, \( y^2 = 16 - 1 = 15 \) and so \( y =  \sqrt{15} \).

    Now that all three sides of the right triangle are known, the definitions can be used to find the other five trigonometric ratios.

    \( \sin \theta = \dfrac{y}{r} = \dfrac{\sqrt{15}}{4} \)          \( \cos \theta = \dfrac{x}{r} = \dfrac{1}{4} \)          \( \tan \theta = \dfrac{y}{x} = \sqrt{15} \)        

     \( \csc \theta = \dfrac{r}{y} = \dfrac{4}{\sqrt{15}} = \dfrac{4 \sqrt{15}}{15} \)          \( \cot \theta = \dfrac{r}{y} = \dfrac{1}{\sqrt{15}} = \dfrac{\sqrt{15}}{15} \)

     Try It  \(\PageIndex{3}\)

    Given \( \tan \theta = \dfrac{\sqrt{3}}{2} \), find the other five trigonometric ratios for angle \( \theta \).

    Answer

    \( \sin \theta = \dfrac{\sqrt{21}}{7} ,  \; \cos \theta = \dfrac{2\sqrt{7}}{7},  \; \csc \theta = \dfrac{\sqrt{21}}{3},   \; \sec \theta = \dfrac{\sqrt{7}}{2},  \; \csc \theta = \dfrac{2\sqrt{3}}{3} \)

    Cofunction Identities

    When working with general right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the sine and cosine of either of the two acute angles in the triangle.

    屏幕快照 2019-07-08 上午6.54.24.png

    Example \(\PageIndex{4}\)

    Using the triangle shown, evaluate \( \sin α, \cos α, \tan α, \sec α, \csc α,\) and \( \cot α\). Also evaluate  sin(\(\beta\)) and cos(\(\beta\)).

    屏幕快照 2019-07-08 上午6.56.19.png

    Solution

    \( \sin α = \dfrac{\text{opposite } α}{\text{hypotenuse}} = \dfrac{4}{5}
    \; \qquad \cos α = \dfrac{\text{adjacent to }α}{\text{hypotenuse}}=\dfrac{3}{5}
    \; \qquad \tan α = \dfrac{\text{opposite }α}{\text{adjacent to }α}=\dfrac{4}{3} \)
    \( \csc α = \dfrac{\text{hypotenuse}}{\text{opposite }α}=\dfrac{5}{4} \qquad \sec α = \dfrac{\text{hypotenuse}}{\text{adjacent to }α}= \dfrac{5}{3} \qquad  \cot α = \dfrac{\text{adjacent to }α}{\text{opposite }α}=\dfrac{3}{4} \)

     

    \(\text{sin}(\beta) = \dfrac{\text{opposite } \beta}{\text{hypotenuse}} = \dfrac{3}{5}\)   \(   \;\;  \text{cos}(\beta) = \dfrac{\text{adjacent to } \beta}{\text{hypotenuse}} = \dfrac{4}{5}   \)

     Try It  \(\PageIndex{4}\)

    A right triangle is drawn with angle \(\alpha\) opposite a side with length 33, angle \(\beta\) opposite a side with length 56, and hypotenuse 65. Evaluate \( \sin α, \cos α,\tan α, \sec α, \csc α,\) and \(\cot α\). Also, find the sine and cosine of \(\beta\)

    Answer

    \( \sin α = \frac{33}{65},  \; \cos α= \frac{56}{65},  \; \tan α= \frac{33}{56},   \; \sec α = \frac{65}{56},  \; \csc α= \frac{65}{33},  \; \cot α= \frac{56}{33} \) \( \qquad \) \(\text{sin} (\beta) = \frac{56}{65},  \; \text{cos} (\beta) = \frac{33}{65}\)

    You may have noticed that in Example \(\PageIndex{4}\) above that \(\cos (\alpha )=\sin (\beta )\) and \(\cos (\beta )=\sin (\alpha )\). This makes sense since the side opposite \(\alpha\) is also adjacent to \(\beta\). Since the three angles in a triangle need to add to \(\pi\), or 180^{\circ}, then the other two angles must add to \(\dfrac{\pi }{2}\), or \(90^{\circ}\), so \(\beta =90^{\circ} -\alpha\), and \(\alpha =90^{\circ} -\beta\). Since \(\cos (\alpha )=\sin (\beta )\), then \(\cos (\alpha )=\sin \left(90^{\circ} -\alpha \right)\).

    Right triangle with angles alpha and beta. Equivalence between sin alpha and cos beta. Equivalence between sin beta and cos alpha.
    Figure \(\PageIndex{4}\): Cofunction identity illustration

    So we may state a cofunction identity: If any two angles are complementary, the sine of one is the cosine of the other, and vice versa. This identity is illustrated in Figure \(\PageIndex{4}\).

    Using this identity, we can state without calculating, for instance, that the sine of \(15^{\circ}\) equals the cosine of \(90^{\circ}- 15^{\circ}\), and that the sine of \(75^{\circ} \) equals the cosine of \(15^{\circ}\). We can also state that if, for a certain angle \(t, \cos t= 75^{\circ} ,\) then \( \sin (90^{\circ}−t)=75^{\circ} \) as well.

    In a similar manner it can be shown that \(\sec (\alpha )=\csc (\beta )\) and \(\sec (\beta )=\csc (\alpha )\). Fiinally, it can be shown that \(\cot (\alpha )=\tan (\beta )\) and \(\cot (\beta )=\tan (\alpha )\). These relationships are summarized below.

    COFUNCTION IDENTITIES

    Cofunctions of complementary angles are equal.

    The angles \(d^{\circ}\) and \((90 ^{\circ}-d^{\circ})\) are complementary angles.  Angles \(r\) and \(\left( \dfrac{π}{2} - r \right)\) are complementary angles
    Sine and cosine are cofunctions. Secant and cosecant are cofunctions. Tangent and cotangent are cofunctions.

    The cofunction identities for an angle \(t\) in degrees are listed below. (If \(t\) is in radians, then use \( \dfrac{π}{2} \) in place of \(90^{\circ}\) .)

    \( \sin t= \cos (90^{\circ}−t)\)  \( \sec t= \csc (90^{\circ}−t) \) \( \tan t= \cot (90^{\circ}−t) \)
    \( \cos t= \sin (90^{\circ}−t)\) \( \csc t= \sec (90^{\circ}−t)\) \( \cot t= \tan (90^{\circ}−t)\)

     How to: Given the sine and cosine of an angle, find the sine or cosine of its complement.

    1. To find the sine of the complementary angle, find the cosine of the original angle.
    2. To find the cosine of the complementary angle, find the sine of the original angle.

    Example \(\PageIndex{5}\): Using Cofunction Identities

    If \( \sin t = \dfrac{5}{12},\) find \( \cos (90^{\circ}−t)\).

    Solution

    According to the cofunction identities for sine and cosine, \( \sin t= \cos (90^{\circ}−t). \) So \( \cos (90^{\circ}−t)= \dfrac{5}{12}. \)

     Try It  \(\PageIndex{5}\)

    If \(\csc (30^{\circ})=2,\) find \( \sec (60^{\circ}).\)

    Answer

    \(2\)

    Given One Side and an Angle, find all other sides and angles

    In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

     How to: Given the side and one acute angle of a right triangle, find the remaining sides.

    1. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
    2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
    3. Using the value of the trigonometric function and the known side length, solve for the missing side length.

    Example \(\PageIndex{6}\): Finding Missing Side Lengths Using Trigonometric Ratios

    Find the unknown sides of the triangle below.

    A right triangle with sides a, c, and 7. Angle of 30 degrees is also labeled which is opposite the side labeled 7.

    Solution

    We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

    \( \tan (30°)= \dfrac{7}{a} \)

    We rearrange to solve for \(a\).  \( a =\dfrac{7}{ \tan (30°)}  =12.1 \)

    We can use the sine to find the hypotenuse. \( \sin (30°)= \dfrac{7}{c} \)

    Again, we rearrange to solve for \(c\). \( c = \dfrac{7}{\sin (30°)} =14 \)

    Notice that if we know at least one of the non-right angles of a right triangle and one side, we can find the rest of the sides and angles.

     Try It  \(\PageIndex{6}\)

    A right triangle has one angle of \(\dfrac{\pi }{3}\) and a hypotenuse of 20. Find the unknown sides and angles of the triangle.

    Answer

    \[\text{cos} \left(\dfrac{\pi}{3} \right) = \dfrac{\text{adjacent}}{\text{hypoteuse}} = \dfrac{\text{Adj}}{20}  \quad \rightarrow \quad
     \text{adjacent} = 20 \text{cos} \left(\dfrac{\pi}{3} \right) = 20 \left(\dfrac{1}{2} \right) = 10\nonumber\]

    \[\text{sin} \left(\dfrac{\pi}{3} \right) = \dfrac{\text{Opposite}}{\text{hypotenuse}} = \dfrac{\text{Opp}}{20} \quad \rightarrow \quad
     \text{opposite} = 20 \text{sin} \left(\dfrac{\pi}{3} \right) = 20 \left(\dfrac{\sqrt{3}}{2} \right) = 10\sqrt{3}\nonumber\]

    Missing angle = 180 - 90 - 60 = 30 degrees or \(\pi / 6\).

    Using Right Triangle Trigonometry to Solve Applied Problems

    Diagram of a radio tower with line segments extending from the top and base of the tower to a point on the ground some distance away. The two lines and the tower form a right triangle. The angle near the top of the tower is the angle of depression. The angle on the ground at a distance from the tower is the angle of elevation.Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's eye.  

    Example \(\PageIndex{7}\)

    To find the height of a tree, a person walks to a point 30 feet from the base of the tree, and measures the angle from the ground to the top of the tree to be 57 degrees. Find the height of the tree.

    Solution

    A tree with angle of 57 degrees from vantage point. Vantage point is 30 feet from tree.We can introduce a variable, \(h\), to represent the height of the tree. The two sides of the triangle that are most important to us are the side opposite the angle, the height of the tree we are looking for, and the adjacent side, the side we are told is 30 feet long.

    The trigonometric function which relates the side opposite of the angle and the side adjacent to the angle is the tangent.

    \(\text{tan} (57^{\circ}) = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{h}{30}\) 

    Solving for \(h\), \(h=30\tan (57{}^\circ )\) 

    Using technology, we can approximate a value \(h=30\tan (57{}^\circ )\approx 46.2\text{ feet}\)

    The tree is approximately 46 feet tall.

    Example \(\PageIndex{8}\)

    A person standing on the roof of a 100 foot tall building is looking towards a skyscraper a few blocks away, wondering how tall it is. She measures the angle of declination from the roof of the building to the base of the skyscraper to be 20 degrees and the angle of inclination to the top of the skyscraper to be 42 degrees.

    Solution

    To approach this problem, it would be good to start with a picture. Although we are interested in the height, 屏幕快照 2019-07-08 上午7.20.05.png\(h\), of the skyscraper, it can be helpful to also label other unknown quantities in the picture – in this case the horizontal distance \(x\) between the buildings and \(a\), the height of the skyscraper above the person.

    To start solving this problem, notice we have two right triangles. In the top triangle, we know one angle is 42 degrees, but we don’t know any of the sides of the triangle, so we don’t yet know enough to work with this triangle.

    In the lower right triangle, we know one angle is 20 degrees, and we know the vertical height measurement of 100 ft. Since we know these two pieces of information, we can solve for the unknown distance \(x\).

    \(\text{tan} (20^{\circ}) = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{100}{x}\)

    Solving for \(x\)

    \(x \text{tan} (20^{\circ}) = 100\)
    \(x = \dfrac{100}{\text{tan} (20^{\circ})}\)

    Now that we have found the distance \(x\), we know enough information to solve the top right triangle.

    \(\text{tan} (42^{\circ}) = \dfrac{\text{opposite}}{\text{adjacent}} = \dfrac{a}{x} = \dfrac{a}{100/\text{tan}(20^{circ})}\)
    \(\text{tan} (42^{\circ}) = \dfrac{a\text{tan} (20^{\circ})}{100}\)
    \(100 \text{tan} (42^{\circ}) = a\text{tan} (20^{\circ})\)
    \(\dfrac{100 \text{tan} (42^{\circ})}{\text{tan} (20^{\circ})} = a\)

    Approximating, \(a=\dfrac{100\tan (42{}^\circ )}{\tan (20{}^\circ )} \approx 247.4\text{ feet}\)

    Adding the height of the first building, we determine that the skyscraper is about 347 feet tall.

    Key Equations

    Cofunction Identities

    \(  \cos t = \sin ( \frac{π}{2}−t) \qquad  \tan t = \cot (\frac{π}{2}−t)  \qquad  \sec t = \csc (\frac{π}{2}−t) \)
    \(  \sin t = \cos (\frac{π}{2}−t)  \qquad  \cot t = \tan (\frac{π}{2}−t)  \qquad  \csc t = \sec (\frac{π}{2}−t) \)

    Key Concepts

    • We can define trigonometric functions as ratios of the side lengths of a right triangle. 
    • The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. 
    • Any two complementary angles could be the two acute angles of a right triangle.
    • If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa.
    • We can use trigonometric functions of an angle to find unknown side lengths.
    • Select the trigonometric function representing the ratio of the unknown side to the known side. 
    • Right-triangle trigonometry permits the measurement of inaccessible heights and distances.
    • The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. 

    Glossary

    adjacent side
    in a right triangle, the side between a given angle and the right angle
    angle of depression
    the angle between the horizontal and the line from the object to the observer’s eye, assuming the object is positioned lower than the observer
    angle of elevation
    the angle between the horizontal and the line from the object to the observer’s eye, assuming the object is positioned higher than the observer
    opposite side
    in a right triangle, the side most distant from a given angle
    hypotenuse
    the side of a right triangle opposite the right angle