Problem 109 - Precalculus

Question

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places. $$ 300^{\circ} $$

Step-by-Step Solution

Answer

Reference angle: 60°, Quadrant: IV, Sine: -0.866, Cosine: 0.500.

Step 1: Find the Reference Angle

To find the reference angle for an angle in standard position, we first determine its coterminal angle between 0° and 360°. Since 300° is already within this range, we use it directly. The reference angle is the acute angle formed with the x-axis, so we subtract 300° from 360°:\[360^{\circ} - 300^{\circ} = 60^{\circ}\]Thus, the reference angle is 60°.

Step 2: Determine the Quadrant

To find the quadrant of the terminal side of 300°, we observe that angles between 270° and 360° are located in the fourth quadrant. Therefore, the terminal side of 300° is in the fourth quadrant.

Step 3: Find the Sine

In the fourth quadrant, the sine value is negative. Using the reference angle of 60° and the fact that for 60° on a unit circle, \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\),we have:\[\sin(300^{\circ}) = -\sin(60^{\circ}) = -\frac{\sqrt{3}}{2}\]Thus, \(\sin(300^{\circ}) = -0.866\).

Step 4: Find the Cosine

In the fourth quadrant, the cosine value is positive. Using the reference angle of 60° and the fact that for 60° on a unit circle, \(\cos(60^{\circ}) = \frac{1}{2}\),we conclude:\[\cos(300^{\circ}) = \cos(60^{\circ}) = \frac{1}{2}\]Thus, \(\cos(300^{\circ}) = 0.500\).

Key Concepts

Reference Angle

When working with trigonometric functions, the reference angle is key to simplifying the calculations. The reference angle is the smallest angle made with the x-axis, helping us analyze angles beyond 90 degrees by bringing them back into the more familiar territory of an acute angle.

• A reference angle is always between 0° and 90°.
• It serves as a bridge to connect different angles to known trigonometric values.
• To find it, determine the coterminal angle between 0° and 360°, if needed, and then subtract from 360° if the angle is in the fourth quadrant.

In our exercise, the original angle is 300°, which lies in the fourth quadrant. We subtract 300° from 360° to get the reference angle of 60°. This transformation makes computations much simpler.

Trigonometric Functions

Trigonometric functions describe relationships between angles and sides in right triangles. They are foundational in understanding geometric properties and real-world applications.

• Sine (\(\sin\) is the ratio of the opposite side to the hypotenuse in a right triangle. It varies from -1 to 1.
• Cosine (
  \(cos\) is the ratio of the adjacent side to the hypotenuse. It also ranges between -1 and 1.

For our angle 300°, using its reference angle of 60°, we know:

• In the unit circle, \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). For 300°, \(\sin(300^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}\), since we are in the fourth quadrant.
• \(\cos(60^\circ) = \frac{1}{2}\). Because the cosine value remains positive in the fourth quadrant, \(\cos(300^\circ) = \cos(60^\circ) = \frac{1}{2}\).

Understanding these functions helps us deduce the sign and value of sine and cosine based on the angle's quadrant position.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane, and it helps visualize and compute trigonometric functions with ease.

• Each point on the circle corresponds to an angle measured from the positive x-axis.
• Trigonometric functions \(\sin\) and \(\cos\) represent the y and x coordinates of these points, respectively.

To determine trigonometric functions for any angle, consider how it maps onto the unit circle. Since the circle is "uni-versally" applicable, understanding it enhances your ability to solve a wide range of problems.
Referring to 300°, locate its position on the unit circle. It's in the fourth quadrant, where \(\sin\) values are negative and \(\cos\) values remain positive. This direct mapping simplifies finding values like \(\sin(300^\circ)\) and \(\cos(300^\circ)\).

Quadrants

The Cartesian plane is divided into four quadrants, each helping us determine the sign of trigonometric functions.

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, while cosine and tangent are negative.
• Quadrant III: Tangent is positive, while sine and cosine are negative.
• Quadrant IV: Cosine is positive, while sine and tangent are negative.

For the angle 300°, the terminal side is in the fourth quadrant. This informs us that the cosine value is positive and the sine value is negative. Understanding the quadrant of an angle is crucial in identifying trigonometric function signs, which in turn aids in accurate computation.