Use the given information to find the quadrant of 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.

Recall the signs of sine and cosine in each quadrant: In quadrant II, sine is positive and cosine is negative; in quadrant I, both sine and cosine are positive.

Find \( \sin s \) using the Pythagorean identity: \( \sin^2 s + \cos^2 s = 1 \). Substitute \( \cos s = -\frac{15}{17} \) to get \( \sin s = +\sqrt{1 - \left(-\frac{15}{17}\right)^2} \) because sine is positive in quadrant II.

Find \( \cos t \) using the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \). Substitute \( \sin t = \frac{4}{5} \) to get \( \cos t = +\sqrt{1 - \left(\frac{4}{5}\right)^2} \) because cosine is positive in quadrant I.

Use the angle addition formulas to find \( \sin(s+t) \) and \( \cos(s+t) \): \[ \sin(s+t) = \sin s \cos t + \cos s \sin t \] \[ \cos(s+t) = \cos s \cos t - \sin s \sin t \] Determine the signs of \( \sin(s+t) \) and \( \cos(s+t) \) to identify the quadrant of \( s + t \).

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Trigonometric Ratios and Signs in Quadrants

Trigonometric functions (sine, cosine, tangent) have specific signs depending 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. Understanding these sign conventions helps determine the values and behavior of angles.

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When adding two angles, the resulting angle's quadrant depends on the sum of their measures. Using the given quadrants and trigonometric values, one can infer the approximate angle measures and determine the quadrant of their sum by considering angle ranges and sign patterns.

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Given one trigonometric ratio, the Pythagorean identity (sin²θ + cos²θ = 1) allows calculation of the other ratio. This is essential when only sine or cosine is provided, enabling complete characterization of the angle and facilitating further calculations like angle addition.

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