Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)

검증된 단계별 안내

Recall that \( \arccot(x) \) is the angle \( \theta \) such that \( \cot(\theta) = x \). Here, we want to find \( \theta = \arccot(\sqrt{3}) \).

Use the definition of cotangent in terms of sine and cosine: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). So, we need to find an angle \( \theta \) where \( \frac{\cos(\theta)}{\sin(\theta)} = \sqrt{3} \).

Recognize that \( \cot(\theta) = \sqrt{3} \) corresponds to a reference triangle where the adjacent side is \( \sqrt{3} \) and the opposite side is 1. This is a 30°-60°-90° triangle, where \( \cot(30^\circ) = \sqrt{3} \).

Determine the quadrant for \( \arccot(\sqrt{3}) \). Since \( \sqrt{3} > 0 \), \( \theta \) lies in the first quadrant where cotangent is positive.

Conclude that \( \arccot(\sqrt{3}) \) is the angle in the first quadrant with reference angle 30°, so \( \theta = 30^\circ \) or in radians \( \theta = \frac{\pi}{6} \).

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주요 개념

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Inverse Trigonometric Functions

Inverse trigonometric functions, like arccot, return the angle whose trigonometric ratio equals a given value. For arccot(√3), we seek an angle whose cotangent is √3. Understanding their ranges and outputs is essential for correctly identifying the angle.

Reference Triangles

Reference triangles are right triangles used to relate trigonometric ratios to specific angles. By constructing a triangle with sides matching the given ratio, we can find the acute reference angle, which helps determine the actual angle in the correct quadrant.

Quadrants and Angle Sign Conventions

The value of trigonometric functions varies by quadrant, affecting the angle's sign and measure. Knowing which quadrant the angle lies in, based on the function and its value, is crucial to correctly interpreting the inverse function's output.

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.8. b. arccot(√3)