Trigonometry and Cosine Study Guide

Updated on Oct 1, 2011

Trigonometry and Cosine

In this lesson, we study a new trigonometric function, cosine. Just as with sine, this function is based on a right triangle. For cosine, however, we use a different pair of sides from that triangle. We find the cosine for the angles of various triangles, and then examine the relationships between sine and cosine.

Cosine is traditionally viewed as the second trigonometric function. The domain of this function consists of all angles x, just like the sine function. We will use angles only between 0° and 90°. This time, when we have a right triangle with angle x, we use the adjacent side A instead of the opposite side. The adjacent side is the leg of the triangle that forms the angle x, together with the hypotenuse, as shown in Figure 6.1.

Figure 6.1

cos(x) =

Example 1

Find the cosine of the angle x in Figure 6.2.

Figure 6.2

Here, the hypotenuse H = √82 and the adjacent side A = 9, so

Example 2

Find the cosine of the angle θ in Figure 6.3.

Figure 6.3

We need to use the Pythagorean theorem to find the length of the adjacent side A.

A2 + 82 = 112
A = √57
cos(θ) =

The Relationship between Sine and Cosine

There are several relationships between the sine and cosine functions. Suppose that a right triangle with angle x has hypotenuse H and legs A and O as depicted in Figure 6.14.

Figure 6.14

sin(x) = and cos(x) =

The third, unmarked angle of this triangle measures 90° – x (or x in radians) because the angles of a triangle must sum to 180° (or π radians). This is illustrated in Figure 6.15.

Figure 6.15

Here, sin(90° – x) = because the side opposite the 90° – x angle is A, the one that is adjacent to angle x. This means that

cos(x) = sin(90° – x)

This is the first of the relationships between sine and cosine.

When two angles add up to 90°, like x and 90° – x, they are called complements. Thus, the cosine of an angle x equals the sine of the complement 90° – x. In fact, the phrase sine of the complement is the origin of the word cosine.

Similarly, cos(90° – x) = = sin(x).

Example 1

If sin(20°) ≈ 0.342, then what is cos(70°)?

Because cos(x) = sin(90° - x), it follows that cos(70°) = sin(90° - 70°) = sin(20°) ≈ 0.342.

The next relationship between sine and cosine comes from the Pythagorean theorem. Start with a right triangle with angle x and sides O, A, and H as shown in Figure 6.16.

Figure 6.16

By the Pythagorean theorem, A2 + O2 = H 2. If we divide both sides of the equation by H2, we get:

(sin(x))2 + (cos(x))2 = 1
sin2(x) + cos2(x) = 1

This is the main relationship between sin (x) and cos(x).


When a trigonometric function like sine or cosine is raised to a power, the exponent is usually put right after the function's name. Thus, sin2(x) = (sin(x))2 and cos5(x) = (cos(x))5.

Example 2

If sin(x) = , then what is cos(x)?

We can solve this by using the formula sin2(x) + cos2(x) = 1.

If we are given the cosine of an angle from a right triangle and any length of the triangle, we can find either of the other sides.

Example 3

If cos(x) = 0.53 for angle x in a right triangle with hypotenuse 10 feet, what are the lengths of the other two sides?

We draw the right triangle, as in Figure 6.17.

Figure 6.17

We know that cos(x) = = 0.53 so the adjacent side is A = 5.3 feet long. We use the Pythagorean theorem to 10 find the third side O.

(5.3)2 + O2 = 102
O = √71.91 ≈ 8.48 feet long

Example 4

Suppose cos(x) = for angle x in a right triangle. If the length of the side opposite x is 12 inches long, what is the hypotenuse of the triangle?

Because cos(x) = , the length O = 12 inches cannot be immediately used. The trick is to use sin(x) = .We must thus figure out what sin(x) is when cos(x) =

sin2(x) + cos2(x) = 1
sin2(x) + = 1
sin(x) =

Thus, sin(x) = , so H = 15. The hypotenuse is 15 inches long.

Practice problems for this study guide can be found at:

Trigonometry and Cosine Practice Questions

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