In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = -2/3, sin θ > 0

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Identify the given information: \(\tan \theta = -\frac{2}{3}\) and \(\sin \theta > 0\). This tells us the tangent ratio and the sign of the sine function, which helps determine the quadrant where \(\theta\) lies.

Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Since \(\tan \theta\) is negative and \(\sin \theta\) is positive, \(\theta\) must be in the second quadrant (where sine is positive and cosine is negative).

Use the Pythagorean identity to find \(\sin \theta\) and \(\cos \theta\). Let the opposite side be 2 and adjacent side be 3 (from the tangent ratio), then the hypotenuse \(r = \sqrt{2^2 + 3^2} = \sqrt{13}\). Since \(\theta\) is in the second quadrant, \(\sin \theta = \frac{2}{\sqrt{13}}\) and \(\cos \theta = -\frac{3}{\sqrt{13}}\).

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

Express all values in exact form, rationalizing denominators if necessary, to find the exact values of all six trigonometric functions for \(\theta\).

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Trigonometric Function Relationships

Trigonometric functions such as sine, cosine, and tangent are interrelated through identities and ratios. Knowing one function's value, like tangent, allows calculation of others using definitions, e.g., tan θ = sin θ / cos θ, which helps find sine and cosine values.

Sign of Trigonometric Functions in Quadrants

The signs of sine, cosine, and tangent depend on the quadrant where the angle θ lies. Since sin θ > 0 and tan θ = -2/3, θ must be in the second quadrant, where sine is positive and tangent is negative, guiding correct sign assignment for other functions.

Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, is essential for finding unknown trigonometric values. After determining sine or cosine from tangent, this identity helps calculate the remaining function values accurately.