Given cos(π/4) = √2/2, determine the exact value of sin(π/8) using the double angle formula.
Understand the Problem
The question is asking to determine the exact value of a trigonometric expression by using the double angle formula.
Answer
The exact value of $ \cos \left(2 \cdot \frac{\pi}{4}\right) $ is $ 0 $.
Answer for screen readers
The exact value of $ \cos \left(2 \cdot \frac{\pi}{4}\right) $ is $ 0 $.
Steps to Solve
- Identify the function and double angle formula
We need to determine the exact value of $ \cos \left(2\theta\right) $ using the double angle formula: $$ \cos(2\theta) = 2\cos^2(\theta) - 1 $$
- Use the given information for $\theta$
Given that $ \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $, we substitute this value into the double angle formula: $$ \cos(2\theta) = 2\left(\frac{\sqrt{2}}{2}\right)^2 - 1 $$
- Calculate the cosine value
Now we calculate $ 2\left(\frac{\sqrt{2}}{2}\right)^2 $:
- First, find $ \left(\frac{\sqrt{2}}{2}\right)^2 $: $$ \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2} $$
- Now substitute back into the double angle formula: $$ \cos(2\theta) = 2 \cdot \frac{1}{2} - 1 = 1 - 1 = 0 $$
The exact value of $ \cos \left(2 \cdot \frac{\pi}{4}\right) $ is $ 0 $.
More Information
The cosine of $ \frac{\pi}{2} $ (which is the result of $ 2 \cdot \frac{\pi}{4} $) is indeed $ 0 $. The double angle formula provides an effective method for determining the cosine value for specific angles.
Tips
- Incorrect squaring: Ensure that when squaring values, calculations are done accurately. For example, squaring $ \frac{\sqrt{2}}{2} $ can sometimes lead to errors if miscalculated.
- Misapplication of the double angle formula: Double angle formulas should be applied precisely; ensure the angles are correctly identified when substituting.
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