Find the exact value of each expression. (Do not use a calculator.)
cos(-15°)

검증된 단계별 안내

Recall the even-odd properties of the cosine function. Specifically, cosine is an even function, which means \(\cos(-\theta) = \cos(\theta)\) for any angle \(\theta\).

Apply this property to the given expression: \(\cos(-15^\circ) = \cos(15^\circ)\).

Express \(15^\circ\) as a difference of two common angles whose cosine and sine values are known, for example, \(15^\circ = 45^\circ - 30^\circ\).

Use the cosine difference identity: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). Substitute \(a = 45^\circ\) and \(b = 30^\circ\) to get \(\cos(15^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ\).

Recall the exact values: \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), and \(\sin 30^\circ = \frac{1}{2}\). Substitute these into the expression to write the exact value of \(\cos(-15^\circ)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Even-Odd Properties of Trigonometric Functions

Cosine is an even function, meaning cos(-θ) = cos(θ). This property allows us to simplify expressions with negative angles by converting them to positive angles without changing the value.

Angle Addition and Subtraction Formulas

These formulas express trigonometric functions of sums or differences of angles, such as cos(a - b) = cos a cos b + sin a sin b. They are essential for finding exact values of angles not commonly found on the unit circle.

Exact Values of Special Angles

Certain angles like 0°, 30°, 45°, 60°, and 90° have known exact sine and cosine values. Using these, along with angle formulas, helps compute exact values for other angles like 15° by expressing them as sums or differences of special angles.

Find the exact value of each expression. (Do not use a calculator.)cos(-15°)