Use the given information to find tan(s + t). See Example 3.
cos s = -15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

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Identify the given information: \( \cos s = -\frac{15}{17} \) with \( s \) in quadrant II, and \( \sin t = \frac{4}{5} \) with \( t \) in quadrant I.

Find \( \sin s \) using the Pythagorean identity \( \sin^2 s + \cos^2 s = 1 \). Since \( s \) is in quadrant II, \( \sin s \) is positive. Calculate \( \sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(-\frac{15}{17}\right)^2} \).

Find \( \cos t \) using the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). Since \( t \) is in quadrant I, \( \cos t \) is positive. Calculate \( \cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - \left(\frac{4}{5}\right)^2} \).

Use the angle addition formula for tangent: \[ \tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t} \]. To apply this, find \( \tan s = \frac{\sin s}{\cos s} \) and \( \tan t = \frac{\sin t}{\cos t} \).

Substitute the values of \( \tan s \) and \( \tan t \) into the formula and simplify the expression to find \( \tan(s + t) \).

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The tangent of the sum of two angles, tan(s + t), can be found using the identity tan(s + t) = (tan s + tan t) / (1 - tan s * tan t). This formula allows combining individual tangent values to find the tangent of their sum.

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Given cos s and sin t along with their quadrants, you can find the missing sine or cosine values using the Pythagorean identity sin²θ + cos²θ = 1. The quadrant information helps determine the sign of these values, which is crucial for accurate tangent calculation.

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The signs of sine, cosine, and tangent depend on the quadrant of the angle. In quadrant II, sine is positive and cosine is negative; in quadrant I, both sine and cosine are positive. Correctly assigning signs ensures the correct evaluation of tan(s + t).

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