Difficulty: Easy
Correct Answer: √3/2
Explanation:
Introduction / Context:
This question tests your understanding of trigonometric functions for negative angles and the symmetry properties of the sine function. Knowing how sine behaves with negative arguments is useful for simplifying trigonometric expressions and solving equations.
Given Data / Assumptions:
Concept / Approach:
First, we use the identity sin(−θ) = −sin θ. So sin(−300°) = −sin(300°). Then we evaluate sin(300°) by finding its reference angle and quadrant. The angle 300 degrees is in the fourth quadrant, and its reference angle is 60 degrees. Sine in the fourth quadrant is negative.
Step-by-Step Solution:
Step 1: Use the odd function property: sin(−300°) = −sin(300°).Step 2: Consider sin(300°). The angle 300° is 360° − 60°, so the reference angle is 60°.Step 3: The sine of 60 degrees is sin 60° = √3/2.Step 4: Because 300° lies in the fourth quadrant, where sine is negative, sin(300°) = −√3/2.Step 5: Substitute into Step 1: sin(−300°) = −[sin(300°)] = −(−√3/2) = √3/2.
Verification / Alternative check:
You can confirm with the unit circle. The point corresponding to 300° has coordinates (1/2, −√3/2). The y coordinate is the sine value, so sin(300°) = −√3/2. Reflecting this angle across the origin to −300° flips the sign of the y coordinate, giving √3/2, which confirms our calculation.
Why Other Options Are Wrong:
- 1 corresponds to sin 90°, not to sin(−300°).
- 1/2 corresponds to sin 30° or sin 150°, not the required angle.
- −√3/2 is the value of sin(300°), not sin(−300°).
- 0 would correspond to integer multiples of 180°, such as 0°, 180°, or 360°, which is not the case here.
Common Pitfalls:
Students sometimes forget that sine is an odd function and assume sin(−θ) = sin θ, which is incorrect. Others confuse the signs of trigonometric functions in different quadrants. Remember the sign rules for quadrants and the basic symmetry property sin(−θ) = −sin θ to avoid these errors.
Final Answer:
The value of sin(−300°) is √3/2.
Discussion & Comments