Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
1. c. tan^(-1)(1/√3)

검증된 단계별 안내

Identify the value inside the inverse tangent function: here it is \(\frac{1}{\sqrt{3}}\).

Recall that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\) in a right triangle, so we want to find an angle \(\theta\) such that \(\tan \theta = \frac{1}{\sqrt{3}}\).

Recognize the common special angle where \(\tan \theta = \frac{1}{\sqrt{3}}\) is \(\theta = 30^\circ\) or \(\theta = \frac{\pi}{6}\) radians, based on the reference triangle with sides 1 (opposite), \(\sqrt{3}\) (adjacent), and 2 (hypotenuse).

Since the problem asks to use reference triangles in an appropriate quadrant, consider the principal value range of \(\tan^{-1}\), which is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so the angle is in the first quadrant where tangent is positive.

Conclude that the angle corresponding to \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\) is the reference angle \(\frac{\pi}{6}\) radians (or \(30^\circ\)) in the first quadrant.

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주요 개념

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

Inverse Tangent Function (arctan)

The inverse tangent function, arctan or tan⁻¹, returns the angle whose tangent is a given value. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known. The output angle is typically in the range (-π/2, π/2) or (-90°, 90°).

Reference Triangles

Reference triangles are right triangles drawn in a coordinate plane to help find angles and trigonometric values in different quadrants. By using the acute angle in the triangle and the signs of trigonometric functions in each quadrant, one can determine the actual angle measure.

Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each affecting the sign of sine, cosine, and tangent functions. Knowing the quadrant helps determine the correct angle corresponding to a trigonometric value, especially when using inverse functions that have restricted ranges.

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.1. c. tan^(-1)(1/√3)