**Trigonometry and Cosine **

**I**n 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.

cos(*x*) =

**Example 1**

Find the cosine of the angle *x* in 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.

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

A^{2}+ 8^{2}= 11^{2}

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.

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.

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.

By the Pythagorean theorem, *A*^{2} + *O*^{2} = *H* ^{2}. If we divide both sides of the equation by H^{2}, we get:

(sin(x))^{2}+ (cos(x))^{2}= 1

sin^{2}(x) + cos^{2}(x) = 1

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

**Tip**

**Tip**

*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, sin ^{2}(x) = (sin(x))^{2} and cos^{5}(x) = (cos(x))^{5}.*

**Example 2**

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

We can solve this by using the formula sin^{2}(x) + cos^{2}(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.

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}+ O^{2}= 10^{2}

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*) =

sin^{2}(x) + cos^{2}(x) = 1

sin^{2}(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:

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