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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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 the Pythagorean identity to find \( \cos s \) and \( \cos t \). Since \( \sin^2 \theta + \cos^2 \theta = 1 \), calculate \( \cos s = \sqrt{1 - \sin^2 s} \) and \( \cos t = -\sqrt{1 - \sin^2 t} \) (negative because \( t \) is in quadrant III where cosine is negative).

Calculate \( \cos s \) explicitly as \( \cos s = \sqrt{1 - \left(\frac{3}{5}\right)^2} \) and \( \cos t = -\sqrt{1 - \left(-\frac{12}{13}\right)^2} \).

Use the sine addition formula: \( \sin(s + t) = \sin s \cos t + \cos s \sin t \). Substitute the known values of \( \sin s, \cos s, \sin t, \cos t \) into this formula.

Simplify the expression by performing the multiplications and additions to express \( \sin(s + t) \) in terms of fractions, without calculating the final numeric value.

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Key Concepts

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

Sine and Cosine Values in Different Quadrants

The signs of sine and cosine functions depend on the quadrant of the angle. In quadrant I, both sine and cosine are positive, while in quadrant III, sine and cosine are both negative. This helps determine the correct values of cosine when only sine is given.

Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This allows us to find the cosine value if the sine value is known, by rearranging to cosθ = ±√(1 - sin²θ), with the sign determined by the quadrant.

Sine Addition Formula

The sine addition formula is sin(s + t) = sin s cos t + cos s sin t. It expresses the sine of a sum of two angles in terms of the sines and cosines of the individual angles, enabling calculation of sin(s + t) from known sine and cosine values.

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