Find the remaining five trigonometric functions of θ.
cos θ = -1/4, sin θ > 0

검증된 단계별 안내

Identify the quadrant where the angle \( \theta \) lies based on the given information: \( \cos \theta = -\frac{1}{4} \) and \( \sin \theta > 0 \). Since cosine is negative and sine is positive, \( \theta \) is in the second quadrant.

Use the Pythagorean identity to find \( \sin \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \]. Substitute \( \cos \theta = -\frac{1}{4} \) to get \[ \sin^2 \theta + \left(-\frac{1}{4}\right)^2 = 1 \].

Solve for \( \sin \theta \) by isolating \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \left(-\frac{1}{4}\right)^2 = 1 - \frac{1}{16} \]. Then take the square root, remembering that \( \sin \theta > 0 \) in the second quadrant.

Calculate the remaining trigonometric functions using the definitions: \[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \csc \theta = \frac{1}{\sin \theta} \].

Substitute the values of \( \sin \theta \) and \( \cos \theta \) into these formulas to express \( \tan \theta \), \( \cot \theta \), \( \sec \theta \), and \( \csc \theta \) in terms of known quantities.

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주요 개념

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

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Given cos θ, this identity allows you to find sin θ by rearranging the equation. Since sin θ > 0, you select the positive root when solving for sin θ.

Signs of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle θ lies. Since cos θ = -1/4 (negative) and sin θ > 0 (positive), θ is in the second quadrant, where sine is positive and cosine is negative. This helps determine the correct signs for all functions.

Definitions of the Six Trigonometric Functions

The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Knowing cos θ and sin θ allows you to calculate tangent (sin θ/cos θ), cotangent (cos θ/sin θ), secant (1/cos θ), and cosecant (1/sin θ). These definitions are essential to find all remaining functions.

Find the remaining five trigonometric functions of θ.cos θ = -1/4, sin θ > 0