Secant, Cosecant, and Cotangent Study Guide

Secant, Cosecant, and Cotangent

In this lesson, we introduce the last three trigonometric functions: secant, cosecant, and cotangent. Again, these are evaluated by dividing one length of a right triangle by another. We then examine the relationships among all six trigonometric functions. If we know the value of one trigonometric function, we will be able to figure out the values of the other five.

Suppose x is the angle of the right triangle in Figure 8.1. The length of the hypotenuse is H, the side opposite x is O, and the adjacent side has length A.

There are only six possible ratios that can be formed by a pair of these sides:

The first three of these have been covered already:

Just for the sake of completeness, we define three more trigonometric functions for the last three: cotangent, secant, and cosecant.

cot(x) =

sec(x) =

csc(x) =

Example

What is the secant, cosecant, and cotangent of the angle x in Figure 8.2?

Here, H = 2√17, 0 = 8, and A = 2, so

Formulas for Secant, Cosecant, and Cotangent

In Lesson 6, we saw that the cosine of an angle is the sine of the complementary angle. This can be written as:

cos(x) = sin(90° – x)

The same is true for the other cofunctions: cotangent and cosecant. To see this, look at the angle x and its complement 90° – x in Figure 8.4.

For angle x, the opposite side has length C and the adjacent side has length B. Thus,

For angle 90° – x, the opposite side has length B and the adjacent side has length C. Thus,

It follows that

cot(x) = tan(90° – x)

and

csc(x) = sec(90° – x)

These properties explain the origins of the names for cosecant and cotangent. Cosecant means "the secant of the complementary angle": and cotangent means "the tangent of the complementary angle."

Similarly, tan(x) = cot(90° – x) and sec(x) = csc(90°– x). (Remember that 90° = – radians, so each 90° – x should be replaced with – x if the angle x is measured in radians.)

While all of these formulas are quite lovely, there are others that are more basic and useful. Take the angle x from the right triangle in Figure 8.5.

The reciprocal (flip) of cos(x) is

Similarly,

Finally,

Tip

This is how most people remember the last three trigonometric functions:

Remember from Lesson 7 that tan(x) = This means that cotangent is also

Thus, every trigonometric function can be written entirely in terms of sin(x) and cos(x).

Example 1

How can csc(x) · tan(x) be represented entirely in terms of sin(x) and cos(x)?

This is, of course, just sec(x), but sometimes is easier to understand.

Example 2

What is in terms of sin(x) and cos(x)?

Example 3

Represent cot(90° – x) · sec(x) in terms of sin (x) and cos(x). Here we first use cot(90° – x) = tan(x), so

More Formulas

In Lesson 6, we used the Pythagorean theorem to prove that

sin²(x) + cos²(x) = 1

If we divide both sides of this equation by cos²(x), we get

tan²(x) + 1 = sec²(x)

Similarly, if we divide both sides of sin²(x) + cos²(x) = 1 by sin²(x), we get

1 + cot²(x) = csc²(x)

With these formulas, we can find one trigonometric value of an angle x given another one.