Use the given information to find sin(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 Pythagorean identity to find \(\sin s\): since \(\sin^2 s + \cos^2 s = 1\), calculate \(\sin s = \pm \sqrt{1 - \cos^2 s}\). Because \(s\) is in quadrant II, where sine is positive, choose the positive root.

Similarly, find \(\cos t\) using \(\sin^2 t + \cos^2 t = 1\): calculate \(\cos t = \pm \sqrt{1 - \sin^2 t}\). Since \(t\) is in quadrant I, where cosine is positive, choose the positive root.

Use the sine addition formula: \(\sin(s + t) = \sin s \cos t + \cos s \sin t\).

Substitute the values of \(\sin s\), \(\cos t\), \(\cos s\), and \(\sin t\) into the formula and simplify to express \(\sin(s + t)\).

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Trigonometric Angle Sum Identity

The sine of the sum of two angles, sin(s + t), can be found using the identity sin(s + t) = sin s cos t + cos s sin t. This formula allows us to express the sine of a combined angle in terms of the sines and cosines of the individual angles.

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Verifying Identities with Sum and Difference Formulas

Determining Sine and Cosine Values from Quadrants

The signs of sine and cosine 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. This helps assign correct signs to trigonometric values when only one ratio is given.

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Sine, Cosine, & Tangent of 30°, 45°, & 60°

Using the Pythagorean Identity to Find Missing Values

Given one trigonometric ratio, the other can be found using sin²θ + cos²θ = 1. For example, if cos s is known, sin s can be calculated as ±√(1 - cos² s), with the sign determined by the quadrant. This is essential for applying the angle sum identity.

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