**Trigonometry and Tangents**

**I**n this lesson, we study the third trigonometric function, **tangent.** Tangent is much like sine and cosine, but it uses different sides of the right triangle. We evaluate the tangent for a number of angles, and examine the various relationships among sine, cosine, and tangent. We also see how knowing one of the trigonometric functions enables us to figure out the values of the others.

The process for evaluating the tangent of an angle is similar to that of sine and cosine:

Step one: Start with an angle of measure x(between 0° and 90°).Step two: Make a right triangle with angle x, as in Figure 7.1.Step three: Measure the length Oof the side opposite the anglexandA, the side adjacent.Step four: The tangent of the angle xis the ratio of the two sides:

tan(*x*) =

As with the sine and cosine, it is really difficult for human beings to draw triangles with great precision, so it is very hard to calculate the tangent by actually constructing a triangle and measuring its sides. Instead, we exercise our understanding of tangent by calculating it for angles in already known triangles.

**Example 1**

What is the tangent of the angle *x* in Figure 7.2?

The tangent of an angle can only be found from the sides of a right triangle. This triangle happens to be right (even though it is not marked as so), because 3^{2} + 4^{2} = 5^{2}. Thus,

O= 3,A= 4, and tan(x) = = 0.75.

There is an interesting difference between the tangent trigonometric function and sine and cosine. Because the sine and cosine are ratios with *H*, the largest side, on the bottom, they could never be bigger than 1. The tangent, which compares the lengths of the two smaller sides, could be quite large.

**Example 2**

What is the tangent of the angle *x* in Figure 7.3?

O= 4 andA= 1, so tan(x) = = 4.

**Example 3**

Find the tangent of angle *x* in Figure 7.4.

To find the length of the adjacent side A, we need to use the Pythagorean theorem:

A^{2}+ 3^{2}= 17^{2}

A = √280 = 2√70

Thus, tan(x) = .

**The Relationship between Sine, Cosine, and Tangent**

The main relationship between tangent, sine, and cosine of an angle *x* is

which can be seen easily from any right triangle with angle *x*, as shown in Figure 7.15.

**Tip**

**Tip**

*Remember that dividil1gby a fraction is the same as multiplying by its flip (reciprocal). For example. *

*because they both become ten when you multiply by . *

*because multiplication and division undo each other. *

*This is why .*

**Example 1**

What is tan(*x*) if sin(*x*) = and cos(*x*) = ?

tan(x) = =

**Example 2**

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

tan(x) = , so sin(x) = cos(x) · tan(x) =

**SOH-CAH-lOA**

Many students remember the foundations of trigonometry with one word: SOH-CAH-TOA, and the triangle in Figure 7.16.

- SOH stands for sin(

*x*) =

- CAH stands for cos(

*x*) =

- TOA stands for tan(

*x*) =

These three ratios are all interesting properties of the angle *x*. However, if you know anyone of them, then you can figure out the other two.

**Example 1**

Suppose the sine of an angle *x* is sin(*x*) = 0.72. What would the cosine and tangent of this angle be?

All we need to do is draw a right triangle that has side lengths that make sin(*x*) = 0.72 = , like the one in Figure 7.17 (there is no need to draw to scale). We then find the other side by the Pythagorean theorem.

The adjacent sideAisA= √1002 – 722 = √4,816 = 4√301.

Thus, the cosine of the angle is cos(*x*) = = and the tangent is tan(*x*) = =

**Example 2**

What is the sine and cosine of angle *x* if tan(*x*) = ?

The easiest way to make a right triangle with angle *x* so that tan (*x*) = ? is by making the opposite side *O* = 2√5 and the adjacent side *A* = 5, as in Figure 7.18. Any choice of opposite and adjacent side length with the ratio would work; because it is easier not to work with fractions, we have chosen the numerator to be the opposite side and the denominator to be the adjacent side.

The hypotenuse *H* of this triangle is found by the Pythagorean theorem.

*H*

^{2}= 5

^{2}+ (2√5)

^{2}

*H*= √25 + 20 = √45 = 3√5

Thus, sin(*x*) = = and cos(*x*) = .

Practice problems for this study guide can be found at:

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