Use the given information to find tan(s + t). See Example 3.
sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

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Identify the given information: \( \sin s = \frac{3}{5} \) with \( s \) in quadrant I, and \( \sin t = -\frac{12}{13} \) with \( t \) in quadrant III.

Recall the formula for \( \tan(s + t) \): \[ \tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \]

Find \( \cos s \) using the Pythagorean identity \( \sin^2 s + \cos^2 s = 1 \). Since \( s \) is in quadrant I, \( \cos s \) is positive: \[ \cos s = \sqrt{1 - \sin^2 s} = \sqrt{1 - \left(\frac{3}{5}\right)^2} \]

Find \( \cos t \) similarly, noting that \( t \) is in quadrant III where cosine is negative: \[ \cos t = -\sqrt{1 - \sin^2 t} = -\sqrt{1 - \left(-\frac{12}{13}\right)^2} \]

Calculate \( \tan s = \frac{\sin s}{\cos s} \) and \( \tan t = \frac{\sin t}{\cos t} \), then substitute these values into the formula for \( \tan(s + t) \) to express the answer.

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Trigonometric Ratios and Quadrants

Trigonometric ratios like sine, cosine, and tangent relate the angles of a triangle to side lengths. The sign of these ratios depends on the quadrant where the angle lies: in quadrant I, all ratios are positive; in quadrant III, sine and cosine are negative, but tangent is positive. Understanding this helps determine the correct values of cosine and tangent for angles s and t.

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Quadratic Formula

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. Given sin θ, you can find cos θ by rearranging this identity, considering the sign based on the quadrant. This is essential to find missing trigonometric values needed to compute tan(s + t).

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Pythagorean Identities

Tangent Addition Formula

The tangent addition formula expresses tan(s + t) as (tan s + tan t) / (1 - tan s * tan t). To use this, you must first find tan s and tan t from the given sine values and quadrant information. This formula allows combining two angles' tangents into a single value.

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