In Exercises 63–82, use a sketch to find the exact value of each expression. tan [cos⁻¹ (− 1/3)]

검증된 단계별 안내

Recognize that the expression is \( \tan(\cos^{-1}(-\frac{1}{3})) \). This means we need to find the tangent of an angle whose cosine is \( -\frac{1}{3} \).

Let \( \theta = \cos^{-1}(-\frac{1}{3}) \). By definition, \( \cos \theta = -\frac{1}{3} \). We want to find \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Use the Pythagorean identity to find \( \sin \theta \): \( \sin \theta = \pm \sqrt{1 - \cos^2 \theta} = \pm \sqrt{1 - \left(-\frac{1}{3}\right)^2} = \pm \sqrt{1 - \frac{1}{9}} = \pm \sqrt{\frac{8}{9}} = \pm \frac{2\sqrt{2}}{3} \).

Determine the correct sign of \( \sin \theta \) by considering the range of \( \theta = \cos^{-1}(-\frac{1}{3}) \). Since \( \cos \theta \) is negative, \( \theta \) lies in the second quadrant where sine is positive. So, \( \sin \theta = \frac{2\sqrt{2}}{3} \).

Finally, calculate \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = -2\sqrt{2} \).

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주요 개념

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

Inverse Cosine Function (cos⁻¹)

The inverse cosine function, cos⁻¹(x), returns the angle whose cosine is x, typically within the range 0 to π radians. It helps determine the angle when the cosine value is known, which is essential for evaluating expressions involving inverse trigonometric functions.

Right Triangle and Unit Circle Relationships

Using a sketch, one can represent the angle from the inverse cosine on the unit circle or as a right triangle. This visualization helps identify the sides of the triangle, enabling the calculation of other trigonometric ratios like tangent by relating opposite and adjacent sides.

Tangent Function and Its Relation to Sine and Cosine

Tangent of an angle is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). Once the cosine value and corresponding angle are known, sine can be found using the Pythagorean identity, allowing the exact value of tangent to be computed.