In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0

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Identify the given information: \(\tan \theta = \frac{4}{3}\) and \(\cos \theta < 0\). Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Determine the quadrant of \(\theta\) based on the signs of \(\tan \theta\) and \(\cos \theta\). Since \(\tan \theta\) is positive and \(\cos \theta\) is negative, \(\theta\) lies in the second quadrant.

Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{4}{3}\) to express \(\sin \theta\) and \(\cos \theta\) in terms of a common variable. Let \(\cos \theta = x\), then \(\sin \theta = \frac{4}{3} x\).

Apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) by substituting \(\sin \theta = \frac{4}{3} x\) and \(\cos \theta = x\), then solve for \(x\).

Once \(\cos \theta\) and \(\sin \theta\) are found, use these values to find the remaining trigonometric functions: \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\).

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Understanding Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Given tan θ = 4/3, this means the opposite side over adjacent side is 4/3. Knowing one ratio allows calculation of the other trigonometric functions using definitions and relationships.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant where the angle lies. Since cos θ < 0, θ is in either the second or third quadrant. This information helps determine the correct signs of sine, cosine, and other functions when calculating their exact values.

Using the Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, allows finding missing trigonometric values when one function is known. After determining the sides from tan θ, this identity helps calculate sine and cosine values, ensuring the results satisfy the fundamental trigonometric relationship.