**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:

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

and

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

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

If we divide both sides of this equation by cos^{2}(*x*), we get

tan^{2}(*x*) + 1 = sec^{2}(*x*)

Similarly, if we divide both sides of sin^{2}(*x*) + cos^{2}(*x*) = 1 by sin^{2}(*x*), we get

1 + cot^{2}(*x*) = csc^{2}(*x*)

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

**Example 1**

If cot(*x*) = , then what is csc(*x*)?

1 + cot^{2}(*x*) = csc^{2}(*x*)

A more general technique for solving such problems was introduced in Lesson 7:

- Draw a right triangle with angle
*x*.
- Use the known trigonometric value to assign lengths to two sides of the triangle.
- Use the Pythagorean theorem to find the third side.
- Use the triangle to find all the other trigonometric values.

**Example 2**

If cot(*x*) = ,then what are the other five trigonometric values of *x*?

Draw a right triangle with angle *x*. If the side adjacent to angle *x* is *A* = 5 and the side opposite is *O* = 2 then this will ensure that cot(*x*) .See Figure 8.6.

The hypotenuse *H* is found with the Pythagorean theorem:

**Example 3**

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

The angle *x* in Figure 8.7 has cos(*x*) = 0. The side O opposite *x* is found with the Pythagorean theorem:

Thus, csc(*x*) =

Practice problems for this study guide can be found at:

Secant, Cosecant, and Cotangent Practice Questions