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Math  /  Trigonometry

QuestionUse the half-angle formula to find the exact value of cos⁡(15∘)=\cos \left(15^{\circ}\right)= □\square
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Studdy Solution

STEP 1

1. The half-angle formula for cosine is applicable.
2. The angle 15∘ 15^\circ can be expressed as half of a known angle.
3. We will use the half-angle identity for cosine: cos⁡(θ2)=±1+cos⁡(θ)2\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}.

STEP 2

1. Identify the angle θ\theta such that 15∘=θ215^\circ = \frac{\theta}{2}.
2. Use the half-angle formula for cosine.
3. Simplify the expression to find the exact value.

STEP 3

Identify the angle θ\theta such that 15∘=θ215^\circ = \frac{\theta}{2}. We know that 30∘30^\circ is twice 15∘15^\circ, so:
θ=30∘\theta = 30^\circ

STEP 4

Apply the half-angle formula for cosine:
cos⁡(θ2)=±1+cos⁡(θ)2\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
Substitute θ=30∘\theta = 30^\circ:
cos⁡(15∘)=±1+cos⁡(30∘)2\cos(15^\circ) = \pm \sqrt{\frac{1 + \cos(30^\circ)}{2}}

STEP 5

Recall that cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}. Substitute this value into the equation:
cos⁡(15∘)=±1+322\cos(15^\circ) = \pm \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}

STEP 6

Simplify the expression inside the square root:
cos⁡(15∘)=±2+34\cos(15^\circ) = \pm \sqrt{\frac{2 + \sqrt{3}}{4}}
cos⁡(15∘)=±2+32\cos(15^\circ) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}
Since 15∘15^\circ is in the first quadrant, where cosine is positive, we choose the positive value:
cos⁡(15∘)=2+32\cos(15^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2}
The exact value of cos⁡(15∘)\cos(15^\circ) is:
2+32\boxed{\frac{\sqrt{2 + \sqrt{3}}}{2}}

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