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 [latex]t[/latex] is its value at [latex]t[/latex] radians. First, we need to create our right triangle. Figure 1 shows a point on a unit circle of radius 1. If we drop a vertical line segment from the point [latex](x,y)[/latex] to the x–axis, we have a right triangle whose vertical side has length [latex]y[/latex] and whose horizontal side has length [latex]x[/latex]. We can use this right triangle to redefine sine, cosine, and the other trigonometric functions as ratios of the sides of a right triangle.

We know

[latex]cos[/latex] [latex]t[/latex] = [latex]\dfrac{x}{1}[/latex] = [latex]x[/latex]

Likewise, we know

[latex]sin[/latex] [latex]t[/latex] = [latex]\dfrac{y}{1}[/latex] = [latex]y[/latex]

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 [latex](x,y)[/latex] coordinates. To be able to use these ratios freely, we will give the sides more general names: Instead of [latex]x[/latex], we will call the side between the given angle and the right angle the adjacent side to angle [latex]t[/latex]. (Adjacent means “next to.”) Instead of [latex]y[/latex], we will call the side most distant from the given angle the opposite side from angle [latex]t[/latex]. And instead of 1, we will call the side of a right triangle opposite the right angle the hypotenuse. These sides are labeled in Figure 2.

Given a right triangle with an acute angle of [latex]t[/latex],

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.”

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 5. The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

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.

 

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 30​°, 60​°, and 45​°, however, remember that when dealing with right triangles, we are limited to angles between 0​° and 90°.

Suppose we have a 30°, 60​°, 90​° triangle, which can also be described as a [latex]\frac{π}{6}[/latex], [latex]\frac{π}{3}[/latex], [latex]\frac{π}{2}[/latex] triangle. The sides have lengths in the relation [latex]s[/latex], [latex]\sqrt{3}s[/latex], [latex]2s[/latex]. The sides of a 45​°, 45​°, 90​° triangle, which can also be described as a [latex]\frac{π}{4}[/latex], [latex]\frac{π}{4}[/latex], [latex]\frac{π}{2}[/latex] triangle, have lengths in the relation [latex]s[/latex], [latex]s[/latex], [latex]\sqrt{2}s[/latex]. These relations are shown in Figure 8.

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

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 [latex]\frac{π}{6}[/latex] and [latex]\frac{π}{3}[/latex], we see that the sine of [latex]\frac{π}{3}[/latex], namely [latex]\frac{\sqrt{3}}{2}[/latex], is also the cosine of [latex]\frac{π}{6}[/latex], while the sine of [latex]\frac{π}{6}[/latex], namely [latex]\frac{1}{2}[/latex], is also the cosine of [latex]\frac{π}{3}[/latex].

See Figure 9.

This result should not be surprising because, as we see from Figure 9, the side opposite the angle of [latex]\frac{π}{3}[/latex] is also the side adjacent to [latex]\frac{π}{6}[/latex], so [latex]sin[/latex] ([latex]\frac{π}{3}[/latex]) and [latex]cos[/latex] ([latex]\frac{π}{6}[/latex]) are exactly the same ratio of the same two sides, [latex]\sqrt{3}s[/latex] and [latex]2s[/latex]. Similarly, [latex]cos[/latex] ([latex]\frac{π}{3}[/latex]) and [latex]sin[/latex] ([latex]\frac{π}{6}[/latex]) are also the same ratio using the same two sides, [latex]s[/latex] and [latex]2s[/latex].

The interrelationship between the sines and cosines of [latex]\frac{π}{6}[/latex] and [latex]\frac{π}{3}[/latex] also holds for the two acute angles in any right triangle, since in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other. Since the three angles of a triangle add to [latex]π[/latex], and the right angle is [latex]\frac{π}{2}[/latex], the remaining two angles must also add up to [latex]\frac{π}{2}[/latex]. That means that a right triangle can be formed with any two angles that add to [latex]\frac{π}{2}[/latex] —in other words, any two complementary angles. 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 10.

Using this identity, we can state without calculating, for instance, that the sine of [latex]\frac{π}{12}[/latex] equals the cosine of [latex]\frac{5π}{12}[/latex], and that the sine of [latex]\frac{5π}{12}[/latex] equals the cosine of [latex]\frac{π}{12}[/latex]. We can also state that if, for a certain angle [latex]t[/latex], [latex]cos t[/latex] = [latex]\frac{5}{13}[/latex], then [latex]sin[/latex] ([latex]\frac{π}{2}[/latex] − [latex]t[/latex]) = [latex]\frac{5}{13}[/latex] as well.

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.

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 12.

Access these online resources for additional instruction and practice with right triangle trigonometry.

• Finding Trig Functions on Calculator
• Finding Trig Functions Using a Right Triangle
• Relate Trig Functions to Sides of a Right Triangle
• Determine Six Trig Functions from a Triangle
• Determine Length of Right Triangle Side

 

Section Exercises

Verbal

1. For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

2. When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the x– and y–coordinates?

3. The tangent of an angle compares which sides of the right triangle?

4. What is the relationship between the two acute angles in a right triangle?

5. Explain the cofunction identity.

Algebraic

For the following exercises, use cofunctions of complementary angles.

6. [latex]cos[/latex] (34°) = [latex]sin[/latex] (__°)

7. [latex]cos[/latex] ([latex]\frac{π}{3}[/latex]) = [latex]sin[/latex] (___)

8. [latex]csc[/latex] (21°) = [latex]sec[/latex] (___°)

9. [latex]tan[/latex] ([latex]\frac{π}{4}[/latex]) = [latex]cot[/latex] (__)

For the following exercises, find the lengths of the missing sides if side [latex]a[/latex] is opposite angle [latex]A[/latex], side [latex]b[/latex] is opposite angle [latex]B[/latex], and side [latex]c[/latex] is the hypotenuse.

10. [latex]cos B[/latex] = [latex]\frac{4}{5}[/latex], [latex]a[/latex] = 10

11. [latex]sin B[/latex] = [latex]\frac{1}{2}[/latex], [latex]a[/latex] = 20

12. [latex]tan A[/latex] = [latex]\frac{5}{12}[/latex], [latex]b[/latex] = 6 

13. [latex]tan A[/latex] = 100, [latex]b[/latex] = 100

14. [latex]sin B[/latex] = [latex]\frac{1}{\sqrt{3}}[/latex], [latex]a[/latex] = 2

15. [latex]a[/latex] = 5, [latex]∡A[/latex] = 60°

16. [latex]c[/latex] = 12, [latex]∡A[/latex] = 45°

Graphical

For the following exercises, use Figure 15 to evaluate each trigonometric function of angle [latex]A[/latex].

17. [latex]sin A [/latex]

18. [latex]cos A[/latex]

19. [latex]tan A[/latex]

20. [latex]csc A[/latex]

21. [latex]sec A[/latex]

22. [latex]cot A[/latex]

For the following exercises, use Figure 16 to evaluate each trigonometric function of angle [latex]A[/latex].

23. [latex]sin A[/latex]

24. [latex]cos A[/latex]

25. [latex]tan A[/latex]

26. [latex]csc A[/latex]

27. [latex]sec A[/latex]

28. [latex]cot A[/latex]

For the following exercises, solve for the unknown sides of the given triangle.

29.

30.

31.

Extensions

32. Find [latex]x[/latex].

33. Find [latex]x[/latex].

34. Find [latex]x[/latex].

35. Find [latex]x[/latex].

36. A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 36​°, and that the angle of depression to the bottom of the tower is 23​°. How tall is the tower?

37. A radio tower is located 325 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 43​°, and that the angle of depression to the bottom of the tower is 31​°. How tall is the tower?

38. A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 15​°, and that the angle of depression to the bottom of the tower is 2​°. How far is the person from the monument?

39. A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 18​°, and that the angle of depression to the bottom of the monument is 3°. How far is the person from the monument?

40. There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be 40​°. From the same location, the angle of elevation to the top of the antenna is measured to be 43​°. Find the height of the antenna.

41. There is lightning rod on the top of a building. From a location 500 feet from the base of the building, the angle of elevation to the top of the building is measured to be 36​°. From the same location, the angle of elevation to the top of the lightning rod is measured to be 38​°. Find the height of the lightning rod.

Real-World Applications

42. A 33-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building?

43. A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80°. How high does the ladder reach up the side of the building?

44. The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

45. The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

46. Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60°, how far from the base of the tree am I?