Problem 92 - A Graphical Approach to Precalculus with Limits

Question

Use a reference angle to find \(\sin \theta\) and \(\cos \theta\) for the given \(\theta\). $$\theta=-210^{\circ}$$

Step-by-Step Solution

Answer

\( \sin(-210^\circ) = -\frac{1}{2} \) and \( \cos(-210^\circ) = -\frac{\sqrt{3}}{2}.\)

Step 1: Determine the Reference Angle

When dealing with negative angles, it's often helpful to first convert the angle into a positive one by adding full circles (additive multiples of 360°). For \( \theta = -210^\circ \), add 360° to get a coterminal angle: \( -210^\circ + 360^\circ = 150^\circ. \) So, the reference angle is the angle inside the unit circle in the positive direction, which is \( 150^\circ - 180^\circ = 30^\circ.\)

Step 2: Determine the Sign in the Third Quadrant

The angle \( 150^\circ \) puts us in the second quadrant when going counter-clockwise. However, since we originally had \( -210^\circ \), it ends up in the third quadrant after adding 360°. In the third quadrant, sine is negative and cosine is negative.

Step 3: Identify Reference Angle Values

The reference angle \( 30^\circ \) allows us to use known trigonometric values. For \( 30^\circ \), \( \sin 30^\circ = \frac{1}{2} \) and \( \cos 30^\circ = \frac{\sqrt{3}}{2}.\)

Step 4: Apply Sign to Sine and Cosine

Considering these values and the signs from the third quadrant, the sine and cosine of our original angle \( \theta = -210^\circ \) are: \( \sin \theta = -\frac{1}{2} \) and \( \cos \theta = -\frac{\sqrt{3}}{2}.\)

Key Concepts

Reference Angle

In trigonometry, a reference angle helps us to find the sine, cosine, and other trigonometric values of angles that go beyond typical 0° to 90° measurements. It's essentially the smallest angle that the terminal side of a given angle forms with the x-axis, regardless of which quadrant the angle ends up in.
To calculate the reference angle, keep the following steps in mind:

• If the angle is in the first quadrant, the reference angle is the angle itself.
• If the angle is in the second quadrant, subtract the angle from 180°.
• If the angle is in the third quadrant, subtract the angle from 180°, similar to the second quadrant.
• If the angle is in the fourth quadrant, subtract the angle from 360°.

For example, when given an angle of -210°, finding the equivalent positive coterminal angle might initially seem complex. However, by adding 360° to it, we find it equals 150°. Since this equivalent angle is in the third quadrant, we calculate the reference angle as 180° - 150° = 30°. This process is essential for simplifying complex angles into more manageable, standard trigonometric values.

Coterminal Angles

Coterminal angles are angles that share the same terminal side, meaning they end up in the same location on the unit circle. Regardless of how many complete spins you make around the circle, both positive and negative spins achieve this.
Finding coterminal angles involves either adding or subtracting full rotations (360° for degrees or \(2\pi\) for radians) from the given angle:

• To find a positive coterminal angle, you can add 360°.
• For a negative coterminal angle, you would subtract 360°.

In the case of the angle \(-210°\), adding 360° gives us a coterminal angle of 150°. Both \(-210°\) and 150° lead the terminal side to the same position in the unit circle, specifically in the third quadrant when considering their placement from the angle's start.

Unit Circle

The unit circle is a fundamental concept in trigonometry with a radius of one. It simplifies learning about trigonometric functions like sine and cosine due to its straightforward nature in connecting angles to their trigonometric values.
Here's why the unit circle is so crucial:

• Angles are measured in degrees or radians, represented by points on the circle's circumference.
• The positive direction is counter-clockwise and begins from the positive x-axis.
• The circle helps locate which quadrant a given angle resides in, which is key to determining the sign (positive or negative) of sine or cosine.

Regarding \(-210°\), once it is adjusted to its positive coterminal angle (150°), the unit circle reveals the angle's quadrant position, which impacts the sign of its sine and cosine. In the third quadrant, both sine and cosine values are negative. By understanding these signs, one can easily determine sine and cosine for any angle, regardless of its initial measurement.