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

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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