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

Verified step by step guidance

1. <insert step 1> Use the Pythagorean identity to find \( \sin s \). Since \( \cos s = -\frac{15}{17} \) and \( s \) is in quadrant II, \( \sin s \) is positive. The identity is \( \sin^2 s + \cos^2 s = 1 \).
2. <insert step 2> Substitute \( \cos s = -\frac{15}{17} \) into the identity: \( \sin^2 s + \left(-\frac{15}{17}\right)^2 = 1 \). Solve for \( \sin^2 s \).
3. <insert step 3> Calculate \( \sin s \) from \( \sin^2 s \). Since \( s \) is in quadrant II, \( \sin s \) is positive.
4. <insert step 4> Use the Pythagorean identity to find \( \cos t \). Since \( \sin t = \frac{4}{5} \) and \( t \) is in quadrant I, \( \cos t \) is positive. The identity is \( \sin^2 t + \cos^2 t = 1 \).
5. <insert step 5> Use the angle addition formula for sine: \( \sin(s + t) = \sin s \cos t + \cos s \sin t \). Substitute the values of \( \sin s \), \( \cos s \), \( \sin t \), and \( \cos t \) to find \( \sin(s + t) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. One key identity is the sine addition formula, which states that sin(s + t) = sin(s)cos(t) + cos(s)sin(t). Understanding these identities is essential for solving problems involving the sum of angles in trigonometry.

Quadrants and Signs of Trigonometric Functions

The unit circle is divided into four quadrants, each affecting the signs of the trigonometric functions. In quadrant II, sine is positive and cosine is negative, while in quadrant I, both sine and cosine are positive. Knowing the quadrant in which an angle lies helps determine the signs of the sine and cosine values needed for calculations.

Finding Missing Trigonometric Values

To find missing trigonometric values, one can use the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1. Given cos(s) = -15/17, we can find sin(s) by calculating sin²(s) = 1 - cos²(s). Similarly, for sin(t) = 4/5, we can find cos(t) using the same identity, which is crucial for applying the sine addition formula.