Use the given information to find the quadrant of s + t. See Example 3.
cos s = -1/5 and sin t = 3/5, s and t in quadrant II

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Identify the given information: \( \cos s = -\frac{1}{5} \) and \( \sin t = \frac{3}{5} \), with both angles \( s \) and \( t \) in quadrant II.

Recall the signs of sine and cosine in quadrant II: sine is positive and cosine is negative in quadrant II. This confirms the given values are consistent with the quadrant.

Find \( \sin s \) using the Pythagorean identity: \( \sin^2 s + \cos^2 s = 1 \). Substitute \( \cos s = -\frac{1}{5} \) to get \( \sin s = +\sqrt{1 - \left(-\frac{1}{5}\right)^2} \) because sine is positive in quadrant II.

Find \( \cos t \) using the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \). Substitute \( \sin t = \frac{3}{5} \) to get \( \cos t = -\sqrt{1 - \left(\frac{3}{5}\right)^2} \) because cosine is negative in quadrant II.

Use the cosine addition formula to find \( \cos(s + t) \): \(\n\[\n\)\( \cos(s + t) = \cos s \cos t - \sin s \sin t \). \(\n\]\nDetermine\) the sign of \( \cos(s + t) \) to identify the quadrant of \( s + t \) based on the signs of sine and cosine in each quadrant.

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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 II, sine is positive while cosine and tangent are negative. Understanding these sign conventions helps identify the possible values and locations of angles.

Sum of Angles and Quadrant Determination

When adding two angles, the resulting angle's quadrant depends on the sum of their measures. Knowing the individual quadrants of angles s and t allows us to estimate the quadrant of s + t by considering angle ranges and how their sine and cosine values combine.

Using Given Trigonometric Values to Find Angle Properties

Given specific trigonometric values like cos s = -1/5 and sin t = 3/5, and knowing their quadrants, we can deduce other function values (like sin s or cos t) using Pythagorean identities. This helps in applying angle sum formulas and determining the quadrant of s + t.

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