Use the given information to find tan(s + t). See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II

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Identify the given information: \(\cos s = -\frac{1}{5}\) and \(\sin t = \frac{3}{5}\), with both angles \(s\) and \(t\) in quadrant II.

Recall that in quadrant II, sine is positive and cosine is negative. Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin s\) and \(\cos t\).

Calculate \(\sin s\) using \(\sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(-\frac{1}{5}\right)^2}\), and since \(s\) is in quadrant II, \(\sin s\) is positive.

Calculate \(\cos t\) using \(\cos t = -\sqrt{1 - \sin^2 t} = -\sqrt{1 - \left(\frac{3}{5}\right)^2}\), because \(t\) is in quadrant II where cosine is negative.

Use the angle addition formula for tangent: \(\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t}\), where \(\tan s = \frac{\sin s}{\cos s}\) and \(\tan t = \frac{\sin t}{\cos t}\). Substitute the values found to express \(\tan(s + t)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Ratios and Quadrants

Trigonometric ratios (sine, cosine, 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 II, sine is positive, cosine is negative, and tangent is negative, which affects how values are interpreted and calculated.

Sum of Angles Formula for Tangent

The tangent of the sum of two angles s and t is given by tan(s + t) = (tan s + tan t) / (1 - tan s * tan t). This formula allows us to find the tangent of a combined angle using the tangents of individual angles, which can be derived from sine and cosine values.

Finding Missing Trigonometric Values

Given one trigonometric ratio and the quadrant, the other ratios can be found using the Pythagorean identity sin²θ + cos²θ = 1. For example, if cos s is known, sin s can be found by solving the identity and considering the sign based on the quadrant.

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