Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact value. (a) \(\cos 150^{\circ}\) (b) \(\cos \left(-60^{\circ}\right)\)

Step-by-Step Solution

Answer

(a) \(\cos 150^{\circ} = -\frac{\sqrt{3}}{2}\), (b) \(\cos(-60^{\circ}) = \frac{1}{2}\).

Step 1: Identify Reference Angle for Part (a)

First, identify the reference angle for \( 150^{\circ} \). Since \( 150^{\circ} \) is in the second quadrant and it is \( 180^{\circ} - 150^{\circ} \), the reference angle is \( 30^{\circ} \).

Step 2: Use Cosine Identity for Second Quadrant

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

Step 3: Calculate Cosine of 30 Degrees

The cosine of \( 30^{\circ} \) is \( \frac{\sqrt{3}}{2} \).

Step 4: Determine the Exact Value for Part (a)

Using the identity from Step 2, \( \cos 150^{\circ} = -\frac{\sqrt{3}}{2} \).

Step 5: Identify Reference Angle for Part (b)

For \(-60^{\circ}\), note that it is equivalent to moving clockwise from the positive \( x \)-axis, placing it in the fourth quadrant. This makes the reference angle \( 60^{\circ} \).

Step 6: Use Cosine Identity for Fourth Quadrant

In the fourth quadrant, the cosine function is positive. So, \( \cos(-60^{\circ}) = \cos(60^{\circ}) \).

Step 7: Calculate Cosine of 60 Degrees

The cosine of \( 60^{\circ} \) is \( \frac{1}{2} \).

Step 8: Determine the Exact Value for Part (b)

Thus, \( \cos(-60^{\circ}) = \frac{1}{2} \).

Key Concepts

Reference Angle

Trigonometry deals with angles in relation to their position along a circle's circumference. When working with angles greater than 90 degrees or negative angles, we often use a **reference angle** to simplify calculations. A reference angle is the acute angle (an angle less than 90 degrees) that an angle makes with the x-axis.

Here's how you find a reference angle:

• **Quadrant 1**: The reference angle is the same as the original angle.
• **Quadrant 2**: Subtract the angle from 180 degrees.
• **Quadrant 3**: Subtract 180 degrees from the angle.
• **Quadrant 4**: Subtract the angle from 360 degrees.

If you have a negative angle, add 360 degrees to find its positive equivalent. Understanding reference angles makes it easier to evaluate trigonometric functions because they share the same sine or cosine values as their reference angles, taking into account the sign based on their quadrant position.

Quadrants

When analyzing angles, we divide the coordinate plane into four sections, known as quadrants. These quadrants help determine the sign of trigonometric functions.

Each quadrant has specific rules:

• **Quadrant 1 (0° to 90°)**: All trigonometric functions are positive.
• **Quadrant 2 (90° to 180°)**: Sine is positive, and cosine and tangent are negative.
• **Quadrant 3 (180° to 270°)**: Tangent is positive, while sine and cosine are negative.
• **Quadrant 4 (270° to 360°)**: Cosine is positive, whereas sine and tangent are negative.

By identifying the quadrant, you can easily determine the signs of the trigonometric functions, simplifying calculations of expressions like cosine, even when the original angle is not within the 0° to 90° range.

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and calculating expressions involving trigonometric functions. These identities express trigonometric functions in terms of others and reveal relationships between them.

Some significant identities include:

• **Pythagorean Identities**: These show the innate relationship between sine, cosine, and tangent, such as \ \( \ \sin^2\theta + \cos^2\theta = 1 \ \).
• **Co-function Identities**: These express functions based on complementary angles, such as \ \( \ \sin(90° - \theta) = \cos\theta\ \ \).
• **Even-Odd Identities**: These help when dealing with negative angles, noting that cosine is an even function \ \( \ \cos(-\theta) = \cos\theta \) , and sine is odd \ \( \ \sin(-\theta) = -\sin\theta \ \).

Using these identities makes solving angle-related problems more manageable, as they reveal consistent patterns and behaviors across all angles. Recognizing these identities not only aids in precise calculations but also enhances problem-solving efficiency in trigonometry.