5.4: Right Triangle Trigonometry
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Learning Objectives
- Use right triangles to evaluate trigonometric functions.
- Find function values for 30°(
),45°( ),and 60°( ). - Use equal cofunctions of complementary angles.
- Use the definitions of trigonometric functions of any angle.
- Use right-triangle trigonometry to solve applied problems.
Mt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains.
We have previously defined the sine and cosine of an angle in terms of the coordinates of a point on the unit circle intersected by the terminal side of the angle:
In this section, we will see another way to define trigonometric functions using properties of right triangles.
Using Right Triangles to Evaluate Trigonometric Functions
In earlier sections, we used a unit circle to define the trigonometric functions. In this section, we will extend those definitions so that we can apply them to right triangles. The value of the sine or cosine function of

We know
Likewise, we know
These ratios still apply to the sides of a right triangle when no unit circle is involved and when the triangle is not in standard position and is not being graphed using

Understanding Right Triangle Relationships
Given a right triangle with an acute angle of
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.”
how to: Given the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that angle
- Find the sine as the ratio of the opposite side to the hypotenuse.
- Find the cosine as the ratio of the adjacent side to the hypotenuse.
- Find the tangent is the ratio of the opposite side to the adjacent side.
Example
Given the triangle shown in Figure

Solution
The side adjacent to the angle is 15, and the hypotenuse of the triangle is 17, so via Equation
Exercise
Given the triangle shown in Figure

- Answer
-
Relating Angles and Their Functions
When working with right triangles, the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in Figure

We will be asked to find all six trigonometric functions for a given angle in a triangle. Our strategy is to find the sine, cosine, and tangent of the angles first. Then, we can find the other trigonometric functions easily because we know that the reciprocal of sine is cosecant, the reciprocal of cosine is secant, and the reciprocal of tangent is cotangent.
how to: Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles
- If needed, draw the right triangle and label the angle provided.
- Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
- Find the required function:
- sine as the ratio of the opposite side to the hypotenuse
- cosine as the ratio of the adjacent side to the hypotenuse
- tangent as the ratio of the opposite side to the adjacent side
- secant as the ratio of the hypotenuse to the adjacent side
- cosecant as the ratio of the hypotenuse to the opposite side
- cotangent as the ratio of the adjacent side to the opposite side
Example
Using the triangle shown in Figure

Solution
Exercise
Using the triangle shown in Figure

- Answer
-
Finding Trigonometric Functions of Special Angles Using Side Lengths
We have already discussed the trigonometric functions as they relate to the special angles on the unit circle. Now, we can use those relationships to evaluate triangles that contain those special angles. We do this because when we evaluate the special angles in trigonometric functions, they have relatively friendly values, values that contain either no or just one square root in the ratio. Therefore, these are the angles often used in math and science problems. We will use multiples of
Suppose we have a

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.
Given trigonometric functions of a special angle, evaluate using side lengths.
- Use the side lengths shown in Figure
for the special angle you wish to evaluate. - Use the ratio of side lengths appropriate to the function you wish to evaluate.
Example
Find the exact value of the trigonometric functions of
Solution
Exercise
Find the exact value of the trigonometric functions of
- Answer
-
Using Equal Cofunction of Complements
If we look more closely at the relationship between the sine and cosine of the special angles relative to the unit circle, we will notice a pattern. In a right triangle with angles of

This result should not be surprising because, as we see from Figure
The interrelationship between the sines and cosines of

Using this identity, we can state without calculating, for instance, that the sine of
COFUNCTION IDENTITIES
The cofunction identities in radians are listed in Table
how to: Given the sine and cosine of an angle, find the sine or cosine of its complement.
- To find the sine of the complementary angle, find the cosine of the original angle.
- To find the cosine of the complementary angle, find the sine of the original angle.
Example
If
Solution
According to the cofunction identities for sine and cosine,
So
Exercise
If
Solution
2
Using Trigonometric Functions
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 a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides
- 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.
- Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
- Using the value of the trigonometric function and the known side length, solve for the missing side length.
Example
Find the unknown sides of the triangle in Figure

Solution
We know the angle and the opposite side, so we can use the tangent to find the adjacent side.
We rearrange to solve for
We can use the sine to find the hypotenuse.
Again, we rearrange to solve for
Exercise
A right triangle has one angle of
- Answer
-
missing angle is
Using Right Triangle Trigonometry to Solve Applied Problems
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. See Figure

how to: Given a tall object, measure its height indirectly
- Make a sketch of the problem situation to keep track of known and unknown information.
- Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
- At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
- Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
- Solve the equation for the unknown height.
Example
To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of 57° 57° between a line of sight to the top of the tree and the ground, as shown in Figure

Solution
We know that the angle of elevation is
The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of
The tree is approximately 46 feet tall.
Exercise
How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of
- Answer
-
About 52 ft
media:
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Key Equations
Cofunction Identities
Key Concepts
- We can define trigonometric functions as ratios of the side lengths of a right triangle. See Example.
- The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See Example.
- We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See Example.
- 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. See Example.
- 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. See Example.
- 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. See Example.
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