Math Problem Statement
Solution
The problem asks you to use mathematical induction to prove the following trigonometric identity:
Step 1: Base Case
For , the left-hand side (LHS) is simply: The right-hand side (RHS) for is: So, the base case holds true.
Step 2: Inductive Hypothesis
Assume the formula holds for , i.e.,
Step 3: Inductive Step
We need to prove that the formula holds for , i.e., that:
Using the inductive hypothesis: [ \sin \theta + \sin 2\theta + \cdots + \sin k\theta + \sin (k+1)\theta = \frac{\sin \frac{k\theta}{2} \sin \frac{(k+1)\theta}{2}}{\
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Mathematical Induction
Summation
Formulas
Summation of sinusoids: sin(θ) + sin(2θ) + ... + sin(nθ) = (sin(nθ/2) * sin((n+1)θ/2)) / sin(θ/2)
Theorems
Principle of Mathematical Induction
Suitable Grade Level
College level or advanced high school (Grades 11-12)
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