Question Find the cosine of -67.5 degrees.

Studdy Solution

STEP 1

1. The cosine function is an even function, which means $\cos(-\theta) = \cos(\theta)$ .
2. The angle $-67.5^{\circ}$ can be expressed as a sum or difference of angles whose cosine values are known.
3. The cosine of $45^{\circ}$ and $22.5^{\circ}$ can be computed using half-angle formulas or known values.

STEP 2

1. Use the property of the cosine function being even to simplify the expression.
2. Express $-67.5^{\circ}$ as a sum or difference of angles.
3. Use the cosine sum/difference formulas to compute the value.
4. Simplify the expression to find the final value.

STEP 3

Use the even property of the cosine function to simplify the expression.
$\cos(-67.5^{\circ}) = \cos(67.5^{\circ})$

STEP 4

Express $67.5^{\circ}$ as a sum or difference of angles whose cosine values are known.
$67.5^{\circ} = 45^{\circ} + 22.5^{\circ}$

STEP 5

Use the cosine sum formula to compute the value:
$\cos(67.5^{\circ}) = \cos(45^{\circ} + 22.5^{\circ})$ $\cos(67.5^{\circ}) = \cos(45^{\circ})\cos(22.5^{\circ}) - \sin(45^{\circ})\sin(22.5^{\circ})$

STEP 6

Compute the cosine and sine of $45^{\circ}$ , which are known values.
$\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$ $\sin(45^{\circ}) = \frac{\sqrt{2}}{2}$

STEP 7

Use the half-angle formula to compute the cosine of $22.5^{\circ}$ .
$\cos(22.5^{\circ}) = \sqrt{\frac{1 + \cos(45^{\circ})}{2}}$ $\cos(22.5^{\circ}) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$

STEP 8

Simplify the expression for $\cos(22.5^{\circ})$ .
$\cos(22.5^{\circ}) = \sqrt{\frac{2 + \sqrt{2}}{4}}$ $\cos(22.5^{\circ}) = \frac{\sqrt{2 + \sqrt{2}}}{2}$

STEP 9

Use the half-angle formula to compute the sine of $22.5^{\circ}$ .
$\sin(22.5^{\circ}) = \sqrt{\frac{1 - \cos(45^{\circ})}{2}}$ $\sin(22.5^{\circ}) = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}$

STEP 10

Simplify the expression for $\sin(22.5^{\circ})$ .
$\sin(22.5^{\circ}) = \sqrt{\frac{2 - \sqrt{2}}{4}}$ $\sin(22.5^{\circ}) = \frac{\sqrt{2 - \sqrt{2}}}{2}$

STEP 11

Substitute the values of $\cos(45^{\circ})$ , $\cos(22.5^{\circ})$ , $\sin(45^{\circ})$ , and $\sin(22.5^{\circ})$ into the sum formula.
$\cos(67.5^{\circ}) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2 + \sqrt{2}}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2 - \sqrt{2}}}{2}\right)$

STEP 12

Simplify the expression to find the final value of $\cos(67.5^{\circ})$ .
$\cos(67.5^{\circ}) = \frac{1}{2}(\sqrt{2 + \sqrt{2}} - \sqrt{2 - \sqrt{2}})$

STEP 13

Conclude that the final value of $\cos(-67.5^{\circ})$ is the same as $\cos(67.5^{\circ})$ .
$\cos(-67.5^{\circ}) = \frac{1}{2}(\sqrt{2 + \sqrt{2}} - \sqrt{2 - \sqrt{2}})$

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