1.7: Trigonometric Functions of Any Angle

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Learning Objectives

Identify reference angles for angles on the unit circle.
Identify the ordered pair on the unit circle for reference angles.
Use ordered pairs on the unit circle to determine trig function values.
Use calculators to find trig function values.

Reference Angles

Reference angles are formed between the terminal side of an angel and the closest part of the $x$-axis.

Consider the angle $150^{\circ}$. If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of $30^{\circ}$, across the $y$−axis.

$Screen Shot 2021-04-16 at 20.04.17.png$ Figure $\PageIndex{1}$

Notice that $150^{\circ}$ makes a $30^{\circ} $ angle with the negative $x$-axis. Therefore we say that $30^{\circ}$ is the reference angle for $150^{\circ}$. Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the $x$-axis. Notice that $30^{\circ}$ is the reference angle for many angles. For example, it is the reference angle for $210^{\circ}$ and for $−30^{\circ}$.

In general, identifying the reference angle for an angle will help you determine the values of the trig functions of the angle.

Identifying Reference Angles

Graph each of the following angles and identify their reference angles.

a. $140^{\circ}$

$140^{\circ} $ makes a $40^{\circ} $ angles with the negative $x$-axis. Therefore the reference angle is $40^{\circ} $.

b. $240^{\circ}$

$240^{\circ}$ makes a $60^{\circ}$ angle with the negative $x$-axis. Therefore the reference angle is $60^{\circ}$

c. $380^{\circ}$

$380^{\circ} $ is a full rotation of $360^{\circ}$, plus an additional $20^{\circ}$. So this angle is co-terminal with $20^{\circ}$, and $20^{\circ}$ is its reference angle.

$Screen Shot 2021-04-16 at 19.42.00.png$ Figure $\PageIndex{2}$

Determining the Value of Trigonometric Functions

1. Find the ordered pair for $240^{\circ}$ and use it to find the value of $\sin 240^{\circ}$.

$\sin 240^{\circ} =−\dfrac{\sqrt{3}}{2}$

As we found in part b under the question above, the reference angle for $240^{\circ}$ is $60^{\circ}$. The figure below shows $60^{\circ}$ and the three other angles in the unit circle that have $60^{\circ}$ as a reference angle.

$Screen Shot 2021-04-16 at 19.43.34.png$ Figure $\PageIndex{3}$

The terminal side of the angle $240^{\circ}$ represents a reflection of the terminal side of $60^{\circ}$ over both axes. So the coordinates of the point are $\left(−\dfrac{1}{2},−\dfrac{\sqrt{3}}{2}\right)$. The y−coordinate is the sine value, so $\sin 240^{\circ} =−\dfrac{\sqrt{3}}{2}$.

Just as the figure above shows $60^{\circ}$ and three related angles, we can make similar graphs for $30^{\circ}$ and $45^{\circ}$.

$Screen Shot 2021-04-16 at 19.44.24.png$ Figure $\PageIndex{4}$

Knowing these ordered pairs will help you find the value of any of the trig functions for these angles.

2. Find the value of $\cot 300^{\circ}$

$\cot 300^{\circ} =−\dfrac{1}{\sqrt{3}}$

Using the graph above, you will find that the ordered pair is $\left(\dfrac{1}{2},−\dfrac{\sqrt{3}}{2}\right)$. Therefore the cotangent value is $\cot 300^{\circ}=\dfrac{x}{y}=\dfrac{\dfrac{1}{2}}{-\dfrac{\sqrt{3}}{2}}=\dfrac{1}{2} \times-\dfrac{2}{\sqrt{3}}=-\dfrac{1}{\sqrt{3}}$

We can also use the concept of a reference angle and the ordered pairs we have identified to determine the values of the trig functions for other angles.

Example $\PageIndex{1}$

Graph $210^{\circ}$ and identify its reference angle.

Solution

The graph of $210^{\circ}$ looks like this:

$Screen Shot 2021-04-16 at 19.46.06.png$ Figure $\PageIndex{5}$

and since the angle makes a $30^{\circ}$ angle with the negative "$x$" axis, the reference angle is $30^{\circ}$.

Example $\PageIndex{2}$

Graph $315^{\circ}$ and identify its reference angle.

Solution

The graph of $315^{\circ}$ looks like this:

$Screen Shot 2021-04-16 at 19.46.16.png$ Figure $\PageIndex{6}$

and since the angle makes a $45^{\circ}$ angle with the positive "$x$" axis, the reference angle is $45^{\circ}$.

Example $\PageIndex{3}$

Find the ordered pair for $150^{\circ}$ and use it to find the value of cos $150^{\circ}$.

Solution

Since the reference angle is $30^{\circ}$, we know that the coordinates for the point on the unit circle are $\left(−\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right)$. This is the same as the value for $30^{\circ}$, except the "$x$" coordinate is negative instead of positive. Knowing this,

$\cos 150^{\circ}=\dfrac{\text { adjacent }}{\text { hypotenuse }}=\dfrac{-\dfrac{\sqrt{3}}{2}}{1}=-\dfrac{\sqrt{3}}{2}$

Review

Graph $100^{\circ}$ and identify its reference angle.
Graph $200^{\circ}$ and identify its reference angle.
Graph $290^{\circ}$ and identify its reference angle.

Calculate each value using the unit circle and special right triangles.

$\sin 225^{\circ}$
$\cos 225^{\circ}$
$\sec 225^{\circ}$

Vocabulary

Term	Definition
Reference Angle	A reference angle is the angle formed between the terminal side of the angle and the closest of either the positive or negative $x$-axis.

Additional Resources

Interactive Element

Video: Reference Angles - Overview

Practice: Reference Angles and Angles in the Unit Circle

Trigonometric Functions of Negative Angles

Recall that graphing a negative angle means rotating clockwise. The graph below shows $−30^{\circ}$.

Notice that this angle is coterminal with $330^{\circ}$. So the ordered pair is $\left(\dfrac{\sqrt{3}}{2},−\dfrac{1}{2} \right)$. We can use this ordered pair to find the values of any of the trig functions of $−30^{\circ}$. For example, $\cos(−30^{\circ})=x=\dfrac{\sqrt{3}}{2}$.

In general, if a negative angle has a reference angle of $30^{\circ}$, $45^{\circ}$, or $60^{\circ}$, or if it is a quadrantal angle, we can find its ordered pair, and so we can determine the values of any of the trig functions of the angle.

Finding the Value of Trigonometric Expressions

Find the value of the following expressions:

1. $\sin(−45^{\circ} )$

$\sin(−45^{\circ} )=−\dfrac{\sqrt{2}}{2}$

$−45^{\circ}$ is in the $4^{th}$ quadrant, and has a reference angle of $45^{\circ}$. That is, this angle is coterminal with $315^{\circ}$. Therefore the ordered pair is $\left(\dfrac{\sqrt{2}}{2},−\dfrac{\sqrt{2}}{2}\right)$ and the sine value is $−\dfrac{\sqrt{2}}{2}$.

2. $\sec(−300^{\circ} )$

$\sec(−300^{\circ} )=2$

The angle $−300^{\circ}$ is in the $1^{st}$ quadrant and has a reference angle of $60^{\circ}$. That is, this angle is coterminal with $60^{\circ}$. Therefore the ordered pair is $\left(\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right)$ and the secant value is $\dfrac{1}{x}=\dfrac{1}{\dfrac{1}{2}}=2$.

3. $\cos(−90^{\circ} )$

$\cos(−90^{\circ} )=0$

The angle $−90^{\circ}$ is coterminal with $270^{\circ}$. Therefore the ordered pair is (0, -1) and the cosine value is 0.

We can also use our knowledge of reference angles and ordered pairs to find the values of trig functions of angles with measure greater than 360 degrees.

Example $\PageIndex{4}$

Earlier, you were asked if it is still possible to find the values of trig functions for the new type of angles.

Solution

What you want to find is the value of the expression: $\cos(−45^{\circ})$

$\cos(−45^{\circ} )=\dfrac{\sqrt{2}}{2}$

Example $\PageIndex{5}$

Find the value of the expression: $\cos −180^{\circ}$

Solution

The angle $−180^{\circ}$ is coterminal with $180^{\circ}$. Therefore the ordered pair of points is $(-1, 0)$. The cosine is the "x" coordinate, so here it is -1.

Example $\PageIndex{6}$

Find the value of the expression: $\sin−90^{\circ}$

Solution

The angle $−90^{\circ}$ is coterminal with $270^{\circ}$. Therefore the ordered pair of points is $(0, -1)$. The sine is the "$y$" coordinte, so here it is -1.

Example $\PageIndex{7}$

Find the value of the expression: $\tan −270^{\circ}$

Solution

The angle $−270^{\circ}$ is coterminal with $90^{\circ}$. Therefore the ordered pair of points is $(0, 1)$. The tangent is the "$y$" coordinate divided by the "$x$" coordinate. Since the "$x$" coordinate is 0, the tangent is undefined.

Review

Calculate each value.

$\sec −120^{\circ}$
$\csc −135^{\circ}$
$\tan −210^{\circ}$
$\sin −270^{\circ}$
$\cot −90^{\circ}$

Vocabulary

Term	Definition
Negative Angle	A negative angle is an angle measured by rotating clockwise (instead of counterclockwise) from the positive $x$ axis.

Additional Resources

Interactive Element

Video: Evaluating Trigonometric Functions of Any Angle - Overview

Practice: Trigonometric Functions of Negative Angles