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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Key Concepts

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

Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each determining the sign of sine, cosine, and tangent functions. In quadrant I, all trigonometric functions are positive, while in quadrant III, sine and cosine are negative but tangent is positive. Knowing the quadrant helps identify the sign of angles and their trigonometric values.

Using Sine Values to Determine Cosine and Other Functions

Given sine values and the quadrant, the Pythagorean identity (sin²θ + cos²θ = 1) allows calculation of cosine values. The sign of cosine depends on the quadrant, which is essential for finding sums or differences of angles using trigonometric identities.

Sum of Angles and Quadrant Determination

To find the quadrant of s + t, use angle sum identities or analyze the signs of sine and cosine of s and t. The sum's quadrant depends on the combined signs of sine and cosine of s + t, which can be deduced from individual values and their quadrants.

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Textbook Question

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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Textbook Question

Verify that each equation is an identity.

(sin 2x)/(sin x) = 2/sec x