Problem 48 - Algebra 2

Question

Find the exact value of each expression. \(\cos 150^{\circ}\)

Step-by-Step Solution

Answer

\(\cos 150^{\circ} = -\frac{\sqrt{3}}{2}\)

Step 1: Identify Reference Angle

The reference angle for 150° is found by subtracting it from 180°, because 150° falls in the second quadrant. So, the reference angle is 180° - 150° = 30°.

Step 2: Determine the Sign of Cosine

In the second quadrant, the cosine of an angle is negative. Therefore, \(\cos(150^{\circ}) = -\cos(30^{\circ})\).

Step 3: Use Known Value of Cosine 30°

The cosine of 30° is a well-known trigonometric value: \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\).

Step 4: Final Calculation

Since \(\cos(150^{\circ}) = -\cos(30^{\circ})\), substitute the known value to get \(\cos(150^{\circ}) = -\frac{\sqrt{3}}{2}\).

Key Concepts

Understanding Reference Angles

A reference angle is the smallest angle that a given angle makes with the horizontal axis. It helps simplify finding the trigonometric values of angles greater than 90 degrees. The concept is essential in trigonometry, as it allows us to express angles in terms of acute angles.
To determine the reference angle, we must first identify the quadrant in which the angle is located. Angles in different quadrants have different ways to find their reference angles:

• 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°.
• For an angle in the third quadrant, subtract it from 180° and take the absolute value of the result.
• If the angle lies in the fourth quadrant, subtract it from 360°.

In our exercise, the angle 150° is in the second quadrant, and thus, its reference angle is 30°, as calculated by 180° - 150°.

Quadrants on the Unit Circle

The coordinate plane, divided into four quadrants, plays a crucial role in trigonometry. Quadrants help determine the sign of trigonometric functions, guiding us in understanding angles better. Here's a brief overview of each quadrant:

• First Quadrant: Both sine and cosine values are positive.
• Second Quadrant: Sine is positive, while cosine is negative. This is because the x-coordinate of a point in this quadrant is negative.
• Third Quadrant: Both sine and cosine are negative, corresponding to both coordinates being negative.
• Fourth Quadrant: Sine is negative, and cosine is positive.

Since 150° is in the second quadrant, the cosine of this angle is negative, validating our solution's step to express \(\cos(150^{\circ}) \) as \(-\cos(30^{\circ}) \).

Exploring Cosine Values

The cosine function provides the ratio of the adjacent side to the hypotenuse in a right triangle. It is crucial to memorize some common cosine values to solve trigonometric problems effectively.
The most frequently used angles in this regard are 0°, 30°, 45°, 60°, and 90°. For these specific angles, here are their cosine values:

• \(\cos(0^{\circ}) = 1\)
• \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)
• \(\cos(45^{\circ}) = \frac{\sqrt{2}}{2}\)
• \(\cos(60^{\circ}) = \frac{1}{2}\)
• \(\cos(90^{\circ}) = 0\)

Given the exercise, we use \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2} \) to find \(\cos(150^{\circ})\) as \(-\frac{\sqrt{3}}{2}\). This usage of the reference angle to utilize known cosine values simplifies calculations and enhances understanding.