Simplify: sin 780° sin 480° + cos 120° sin 30°
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A.
0
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B.
2/3
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C.
1/3
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D.
1/2
Correct Answer:
D. 1/2
Explanation:
To solve the expression sin 780° sin 480° + cos 120° sin 30°, we can simplify each trigonometric term using their periodic properties and reference angles: - sin 780°: Since sine has a period of 360°, sin 780° is equivalent to sin(780° - 2 × 360°) = sin 60°. The value of sin 60° is √3/2. - sin 480°: Similarly, sin 480° is equivalent to sin(480° - 360°) = sin 120°. Using the identity sin(180° - θ) = sin θ, sin 120° = sin 60° = √3/2. - cos 120°: This can be written as cos(180° - 60°). Since cosine is negative in the second quadrant, cos 120° = -cos 60° = -1/2. - sin 30°: The standard value for sin 30° is 1/2.Now, substitute these values back into the original expression: (√3/2 × √3/2) + (-1/2 × 1/2) \= 3/4 - 1/4 \= 2/4 \= 1/2 The simplified value is 1/2, which corresponds to Option 4 (D).
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