Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\sin 92^{\circ}\) (b) \(\sin \left(-92^{\circ}\right)\)

Step-by-Step Solution

Answer

The expressions \(\sin 92^{\circ}\) and \(\sin \left(-92^{\circ}\right)\) can be rewritten in terms of a positive acute angle as \(\cos(2^{\circ})\) and \(-\sin(88^{\circ})\)respectively.

Step 1: Determine The Equivalent Positive Acute Angle of sin(92)

The angle 92° is located in the second quadrant of the unit circle and it's 2° more than 90°. In the second quadrant, if so 92 = 90 + 2 (since it is 2° more than 90° angle). Thus, sin(92°) can be rewritten as sin(90 + 2). The sin(90 + θ) is equal to cos(θ), hence, sin(92°) = sin(90° + 2°) = cos(2°).

Step 2: Determine The Equivalent Positive Acute Angle of sin(-92)

The negative angle signifies that angle is measured in a clockwise direction. When we add 360° to any angle the terminal side of the angle remains the same, and so does the value of the sine. Hence, sin(-92°) gives the same result as sin(-92° + 360°) = sin(268°). The angle 268° is in the third quadrant, and it is 92° more than 180°. In the third quadrant, the sine function has negative of the value they would have at their reference angle. So, 268 = 180 + 88 (since it is 88° more than 180°).Thus, sin (-92°) can be rewritten as sin(180° + 88°) = -sin(88°).

Key Concepts

Reference Angle

A reference angle is incredibly useful when dealing with trigonometric functions and the unit circle. It's essentially the acute angle between the terminal side of the given angle and the x-axis.

• Reference angles are always positive.
• They range from 0° to 90°.

The reference angle helps us simplify trigonometric expressions, by relating any angle to the corresponding acute angle in the first quadrant. For example, in the original exercise, the reference angle for \(92^{\circ}\) is \(2^{\circ}\) because it's 2° beyond the 90°, situated in the second quadrant. Thus, the sine function can readily be calculated or approximated as \(\cos(2^{\circ})\). Remember, when using reference angles:

• Subtract 90° for angles in the second quadrant.
• Subtract 180° for angles in the third quadrant.
• Subtract 270° for angles in the fourth quadrant.

Knowing how to find the reference angle is a key skill in solving trigonometry problems effectively.

Quadrants of the Unit Circle

The unit circle is a fundamental concept in trigonometry, with its quadrants playing a crucial role in evaluating angles.

• Quadrant I: Angles between 0° and 90°, both sine and cosine are positive.
• Quadrant II: Angles between 90° and 180°, sine is positive, cosine is negative.
• Quadrant III: Angles between 180° and 270°, both sine and cosine are negative.
• Quadrant IV: Angles between 270° and 360°, sine is negative, cosine is positive.

Knowing which quadrant an angle lies in helps determine the sign and value of trigonometric functions. For instance, the exercise discusses \(92^{\circ}\) located in the second quadrant, where the sine function is positive, and \(268^{\circ}\) in the third quadrant, where the sine value becomes negative.Understanding the properties of the unit circle's quadrants enables better comprehension of trigonometric functions, as trigonometry tests often utilize knowledge of quadrants and corresponding signs to solve problems.

Positive and Negative Angles

Angles in trigonometry can be positive or negative. Understanding their difference is essential.

• Positive angles are measured counterclockwise from the positive x-axis.
• Negative angles are measured clockwise from the positive x-axis.

In the original exercise, we observe the angle of \(-92^{\circ}\). Since it is negative, it's measured in a clockwise direction. To simplify negative angles, we often add 360° until they become positive. For example, \(-92^{\circ} + 360^{\circ} = 268^{\circ}\) puts the angle into a positive form while retaining equivalent trigonometric values.By converting negative angles into positive ones, you can better utilize the known behaviors of trigonometric functions across the unit circle. This method is vital for solving and verifying trigonometric functions and ensuring correct quadrants and signs are used in your calculations.