In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 8/17, 270° < θ < 360°

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Identify the quadrant where the angle \( \theta \) lies. Since \( 270^\circ < \theta < 360^\circ \), \( \theta \) is in the fourth quadrant, where cosine is positive and sine is negative.

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this to find \( \sin \theta \) by substituting \( \cos \theta = \frac{8}{17} \).

Calculate \( \sin \theta = -\sqrt{1 - \cos^2 \theta} = -\sqrt{1 - \left(\frac{8}{17}\right)^2} \) because sine is negative in the fourth quadrant.

Find \( \tan \theta \) using the definition \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the values of \( \sin \theta \) and \( \cos \theta \) found in previous steps.

Determine the remaining trigonometric functions using their relationships: \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).

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Trigonometric Functions and Their Relationships

Trigonometric functions include sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent. Knowing one function value, such as cosine, allows you to find others using identities and relationships, like sin²θ + cos²θ = 1.

Quadrant Sign Rules

The sign of trigonometric functions depends on the quadrant of the angle. Since 270° < θ < 360° (fourth quadrant), cosine is positive, sine is negative, and tangent is negative. This helps determine the correct sign of each function value.

Using the Pythagorean Identity to Find Sine

Given cos θ, sine can be found using sin²θ = 1 - cos²θ. After calculating the magnitude, the quadrant determines the sign of sine. This step is essential to find all remaining trigonometric functions accurately.