Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

3. Unit Circle

Reference Angles

Trigonometry

3. Unit Circle

Reference Angles

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Problem 81

Textbook Question

Suppose θ is in the interval (90°, 180°). Find the sign of each of the following.sin(-θ)

Verified step by step guidance

Identify the quadrant in which \( \theta \) lies. Since \( \theta \) is in the interval (90^\circ, 180^\circ), it is in the second quadrant.

Recall that the sine function is positive in the second quadrant. Therefore, \( \sin(\theta) \) is positive.

Use the identity for the sine of a negative angle: \( \sin(-\theta) = -\sin(\theta) \).

Since \( \sin(\theta) \) is positive in the second quadrant, \( -\sin(\theta) \) will be negative.

Conclude that \( \sin(-\theta) \) is negative.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Understanding Quadrants

In trigonometry, the unit circle is divided into four quadrants, each corresponding to specific ranges of angles. The first quadrant (0° to 90°) has all positive sine and cosine values, while the second quadrant (90° to 180°) has a positive sine and a negative cosine. Knowing the quadrant in which an angle lies helps determine the signs of the trigonometric functions.

Properties of Sine Function

The sine function is an odd function, meaning that sin(-θ) = -sin(θ). This property is crucial for evaluating sine values for negative angles. Understanding this property allows us to relate the sine of a negative angle to the sine of its positive counterpart, which is essential for solving the given problem.

Angle Reference and Sign Determination

When dealing with angles in trigonometry, it's important to consider the reference angle and its position in the unit circle. For θ in the interval (90°, 180°), the reference angle is 180° - θ. Since sine is positive in the second quadrant, we can conclude that sin(θ) is positive, and thus sin(-θ) will be negative due to the odd function property.