Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Feb 06, 2024

Simplify $\cos(\theta - \pi/6)$ using trigonometric identities.

STEP 1

Assumptions
1. We need to use the cosine difference formula to expand the expression.
2. The cosine difference formula is: $\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$
3. We know the exact values of $\cos(\frac{\pi}{6})$ and $\sin(\frac{\pi}{6})$ .

STEP 2

Write down the cosine difference formula that we will use to expand the expression.
$\cos(\theta - \frac{\pi}{6}) = \cos(\theta)\cos(\frac{\pi}{6}) + \sin(\theta)\sin(\frac{\pi}{6})$

STEP 3

Recall the exact values of $\cos(\frac{\pi}{6})$ and $\sin(\frac{\pi}{6})$ .
$\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{6}) = \frac{1}{2}$

STEP 4

Substitute the known values into the difference formula.
$\cos(\theta - \frac{\pi}{6}) = \cos(\theta)\left(\frac{\sqrt{3}}{2}\right) + \sin(\theta)\left(\frac{1}{2}\right)$

STEP 5

Simplify the expression by distributing the cosine and sine terms.
$\cos(\theta - \frac{\pi}{6}) = \frac{\sqrt{3}}{2}\cos(\theta) + \frac{1}{2}\sin(\theta)$
The expression $\cos(\theta - \frac{\pi}{6})$ is now expanded and simplified.

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