Trigonometry
Simplification of trigonometric expression; weighted sum
MJMT_Full_Test_02
Grade 12
Question:
Let $T(\theta) = \cos^2(30°-\theta) - \cos(30°-\theta)\cos(30°+\theta) + \cos^2(30°+\theta)$. Then the value of $4\displaystyle\sum_{\theta=1}^{30} \theta\, T(\theta)$ is
Step-by-Step Solution
Key Concept: Simplify $T(\theta)$ using sum-to-product identities. Show $T(\theta)$ is a constant (independent of $\theta$).
Simplify: $T(\theta)=\frac{3}{4}$. So $4\sum_{\theta=1}^{30}\theta T(\theta)=3\times\frac{30\times31}{2}=1395$.
Correct Answer: 1