Difficulty: Easy
Correct Answer: 0
Explanation:
Introduction / Context:
This trigonometry question tests your understanding of the parity (odd or even nature) of sine and cosine functions. It focuses on how these functions behave when the angle is negative. Using these basic identities allows you to convert negative angles into equivalent positive ones, which are easier to evaluate from standard trigonometric values.
Given Data / Assumptions:
Concept / Approach:
The sine function is an odd function, meaning sin(−θ) = −sin(θ). The cosine function is an even function, meaning cos(−θ) = cos(θ). By applying these properties, you can rewrite each term with a positive angle. Then you simply substitute the known values for sin(π/3) and cos(π/6) and add the results.
Step-by-Step Solution:
Verification / Alternative check:
You can verify the result numerically by approximating √3 as about 1.732. Then sin(π/3) is about 0.866, so sin(−π/3) is about −0.866. Similarly, cos(π/6) is about 0.866, so cos(−π/6) is also 0.866. Adding these approximate values gives −0.866 + 0.866 ≈ 0, which agrees with the exact symbolic evaluation.
Why Other Options Are Wrong:
The values 1, 2, 3, and −1 would require the two terms either to be equal and positive, equal and negative, or of different magnitudes. Here, the two terms are equal in magnitude and opposite in sign, so they cancel perfectly. Any nonzero result would contradict the symmetry of sine and cosine for these particular angles.
Common Pitfalls:
A common error is to forget which function is odd and which is even, for example writing cos(−θ) = −cos(θ), which is incorrect. Another mistake is to misremember the basic values sin(π/3) and cos(π/6), even though they are identical. Keeping a small table of standard angles and their sine and cosine values in mind is very helpful for such questions.
Final Answer:
The exact value of sin(−π/3) + cos(−π/6) is 0.
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