Problem 20 - Algebra and Trigonometry

Question

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. $$ 1-2 \sin ^{2} \frac{\pi}{12} $$

Step-by-Step Solution

Answer

\sqrt{3}/2

Step 1: Identify the double angle formula

By inspecting the expression \(1-2 \sin^{2}(\pi/12)\), you should recognize that the form of the expression matches the form of one of the cosine double angle formulas \(\cos(2A) = 1 - 2\sin^2(A)\).

Step 2: Apply the identified formula

Substitute \(A=\pi/12\) into the cosine double angle formula, you will get the expression into the form of a cosine double angle. We then rewrite the expression as \(\cos(2 \cdot \pi/12)\). This simplifications gives us \(\cos(\pi/6)\).

Step 3: Calculate the exact value

The next task is to compute the exact value of the expression, which is now \(\cos(\pi/6)\). From the unit circle, or remembering the cosine values at special angles, we know that \(\cos(\pi/6) = \sqrt{3}/2\)

Key Concepts

Understanding Trigonometry

Gaining proficiency in trigonometry involves understanding how to find exact values for trigonometric functions. Exact values are specific numerical values that we get from trigonometric functions at certain angles without any approximation or rounding.

Finding exact values often involves knowing key values on the unit circle, where the radius is one. Notable angles, such as pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{pi/red{(d)red{_a>red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{(a)red{(a)red{_a)red{_a)red{_a)red{_a)red{(a)red{(a)red{_a)red{(a)red{_a)red{_a)red{(a)red{(a)red{(a)red{(a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{_a)red{(a)_a)red{(a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)_a)mrad{rad{rd{rd{rd{20).