Relating Trigonometric Functions

Learning objectives

A student will be able to:

State the reciprocal relationships between trig functions, and use these identities to find values of trig functions.
State quotient relationships between trig functions, and use quotient identities to find values of trig functions.
State the domain and range of each trig function.
State the sign of a trig function, given the quadrant in which an angle lies.
State the Pythagorean identities and use these identities to find values of trig functions.

Video: Trigonometric Identities - Reciprocal, Quotient, Pythagorean

Introduction

In previous lessons we defined and worked with the six trig functions individually. In this lesson, we will consider relationships among the functions. In particular, we will develop several identities involving the trig functions. An identity is an equation that is true for all values of the variables, as long as the expressions or functions involved are defined. For example, $x + x = 2x$ is an identity. In this lesson we will develop several identities involving trig functions. Because of these identities, the same function can have very many different algebraic representations. These identities will allow us to relate the trig functions' domains and ranges, and the identities will be useful in solving problems in later chapters.

Reciprocal identities

The first set of identities we will establish are the reciprocal identities. A reciprocal of a fraction $\frac{a} {b}$ is the fraction $\frac{b} {a}$ . That is, we find the reciprocal of a fraction by interchanging the numerator and the denominator, or flipping the fraction. The six trig functions can be grouped in pairs as reciprocals.

First, consider the definition of the sine function for angles of rotation: $\sin \theta = \frac{y} {r}$ . Now consider the cosecant function: $\csc \theta = \frac{r} {y}$ . In the unit circle, these values are $\sin \theta = \frac{y} {1} = y$ and $\csc \theta = \frac{1} {y}$ . These two functions, by definition, are reciprocals. Therefore the sine value of an angle is always the reciprocal of the cosecant value, and vice versa. For example, if $\sin \theta = \frac{1} {2}$ , then $\csc \theta = \frac{2} {1} = 2$ .

Analogously, the cosine function and the secant function are reciprocals, and the tangent and cotangent function are reciprocals:

$\sec\ \theta = \frac{1} {\cos \theta}\ \text{or}\ \cos \theta = \frac{1} {\sec\ \theta} \ \cot \theta = \frac{1} {\tan \theta}\ \text{or}\ \tan \theta = \frac{1} {\cot \theta}$

We can use these reciprocal relationships to find values of trig functions. The fundamental identity stemming from the Pythagorean Theorem $1 = \sin^2 x + \cos^2 x$ can take a great many new forms.

Example 1: Find the value of each expression using a reciprocal identity.

a. $\cos \theta = .3, \sec\ \theta = ?$

b. $\cot \theta = \frac{4} {3},\cot \theta = ?$

Solution:

a. $\sec\ \theta = \frac{10} {3}$

These functions are reciprocals, so if $\cos \theta = .3$ , then $\sec\ \theta = \frac{1} {.3}$ . It is easier to find the reciprocal if we express the values as fractions: $\cos \theta = .3 = \frac{3} {10} \Rightarrow \sec\ \theta = \frac{10} {3}$ .

b. $\tan \theta = \frac{3} {4}$

These functions are reciprocals, and the reciprocal of $\frac{4} {3}$ is $\frac{3} {4}$ .

We can also use the reciprocal relationships to determine the domain and range of functions.

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