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

Verified step by step guidance

1. <Step 1: Understand the problem. We need to determine the quadrant of the angle s + t given that cos(s) = -1/5 and sin(t) = 3/5, with both angles s and t located in quadrant II.>
2. <Step 2: Recall the properties of trigonometric functions in quadrant II. In quadrant II, cosine is negative and sine is positive. This is consistent with the given values: cos(s) = -1/5 and sin(t) = 3/5.>
3. <Step 3: Determine the signs of sin(s) and cos(t) using the quadrant information. Since s is in quadrant II, sin(s) is positive. Similarly, since t is in quadrant II, cos(t) is negative.>
4. <Step 4: Use the angle addition formulas to find the signs of sin(s + t) and cos(s + t). The formulas are: sin(s + t) = sin(s)cos(t) + cos(s)sin(t) and cos(s + t) = cos(s)cos(t) - sin(s)sin(t).>
5. <Step 5: Analyze the signs of sin(s + t) and cos(s + t) to determine the quadrant of s + t. If sin(s + t) is positive and cos(s + t) is negative, s + t is in quadrant II. If both are negative, s + t is in quadrant III. If sin(s + t) is negative and cos(s + t) is positive, s + t is in quadrant IV.>

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In this context, cos(s) = -1/5 indicates that angle s has a cosine value in the second quadrant, where cosine is negative. Similarly, sin(t) = 3/5 shows that angle t has a sine value in the second quadrant, where sine is positive.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, each defined by the signs of the x (cosine) and y (sine) coordinates. Quadrant I has both positive sine and cosine, Quadrant II has a negative cosine and positive sine, Quadrant III has both negative sine and cosine, and Quadrant IV has a positive cosine and negative sine. Understanding these quadrants is essential for determining the signs of trigonometric functions based on the angles.

Sum of Angles in Trigonometry

The sum of angles in trigonometry involves finding the sine and cosine of the sum of two angles, s and t, using the formulas: sin(s + t) = sin(s)cos(t) + cos(s)sin(t) and cos(s + t) = cos(s)cos(t) - sin(s)sin(t). To find the quadrant of s + t, one must evaluate the signs of these resulting sine and cosine values, which will indicate the quadrant in which the angle s + t lies.