Use the given information to find the quadrant of s + t. See Example 3.
sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

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Identify the given information: \( \sin s = \frac{3}{5} \) with \( s \) in quadrant I, and \( \sin t = -\frac{12}{13} \) with \( t \) in quadrant III.

Recall that in quadrant I, all trigonometric functions are positive, and in quadrant III, sine is negative while cosine is also negative.

Find \( \cos s \) using the Pythagorean identity: \( \cos s = \sqrt{1 - \sin^2 s} = \sqrt{1 - \left(\frac{3}{5}\right)^2} \). Since \( s \) is in quadrant I, \( \cos s \) is positive.

Find \( \cos t \) similarly: \( \cos t = -\sqrt{1 - \sin^2 t} = -\sqrt{1 - \left(-\frac{12}{13}\right)^2} \) because \( t \) is in quadrant III, where cosine is negative.

Use the cosine addition formula to find \( \cos(s + t) \): \[ \cos(s + t) = \cos s \cos t - \sin s \sin t \]. Then determine the sign of \( \cos(s + t) \) and \( \sin(s + t) \) to identify the quadrant of \( s + t \).

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Pratica correlata

Domanda del libro di testo

Use the given information to find sin(s + t). See Example 3.

sin s = 3/5 and sin t = -12/13, s in quadrant I and t in quadrant III

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Domanda del libro di testo

Verify that each equation is an identity.