In Exercises 41–43, find the exact value of each of the remaining trigonometric functions of θ.
cos θ = 2/5, sin θ < 0

검증된 단계별 안내

Identify the given information: \(\cos \theta = \frac{2}{5}\) and \(\sin \theta < 0\). This tells us the cosine value and that the sine is negative, which helps determine the quadrant where \(\theta\) lies.

Recall the Pythagorean identity: \(\sin^{2} \theta + \cos^{2} \theta = 1\). Use this to find \(\sin \theta\) by substituting \(\cos \theta = \frac{2}{5}\) into the equation.

Calculate \(\sin^{2} \theta = 1 - \cos^{2} \theta = 1 - \left(\frac{2}{5}\right)^{2} = 1 - \frac{4}{25} = \frac{21}{25}\). Then, \(\sin \theta = \pm \sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5}\).

Since \(\sin \theta < 0\), choose the negative value: \(\sin \theta = -\frac{\sqrt{21}}{5}\).

Find the remaining trigonometric functions using the definitions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) to express each function exactly.

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주요 개념

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Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. Given cos θ, this identity allows you to find sin θ by rearranging the equation. Since cos θ = 2/5, you can calculate sin θ as ±√(1 - (2/5)²), choosing the sign based on the given condition.

Sign of Trigonometric Functions in Quadrants

The sign of sine and cosine depends on the quadrant where the angle θ lies. Since sin θ < 0 and cos θ = 2/5 (positive), θ must be in the fourth quadrant, where cosine is positive and sine is negative. This helps determine the correct sign for sin θ and other functions.

Definition of Trigonometric Functions

Trigonometric functions such as tangent, cotangent, secant, and cosecant are defined in terms of sine and cosine. For example, tan θ = sin θ / cos θ, sec θ = 1 / cos θ, etc. Once sin θ and cos θ are known, these remaining functions can be calculated exactly.

In Exercises 41–43, find the exact value of each of the remaining trigonometric functions of θ.cos θ = 2/5, sin θ < 0