Question

Use the given information to find the quadrant of s + t. See Example 3.cos s = - 15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Accepted Answer

Identify the given information: $$ \cos s = -\frac{15}{17} $$ with $$ s  in $$quadrant II, and $$ \sin t = \frac{4}{5} $$ with $$ t  in $$quadrant I. Recall the signs of sine and cosine in each quadrant: In quadrant II, sine is positive and cosine is negative; in quadrant I, both sine and cosine are positive. Find $$ \sin s $$ using the Pythagorean identity: $$ \sin^2 s + \cos^2 s = 1 . $$Substitute $$ \cos s = -\frac{15}{17}  to $$get $$ \sin s = +\sqrt{1 - \left(-\frac{15}{17}\right)^2} $$ because sine is positive in quadrant II. Find $$ \cos t $$ using the Pythagorean identity: $$ \sin^2 t + \cos^2 t = 1 . $$Substitute $$ \sin t = \frac{4}{5}  to $$get $$ \cos t = +\sqrt{1 - \left(\frac{4}{5}\right)^2} $$ because cosine is positive in quadrant I. Use the angle addition formulas to find $$ \sin(s+t) $$ and $$ \cos(s+t) $$:

$$ \sin(s+t) = \sin s \cos t + \cos s \sin t $$

$$ \cos(s+t) = \cos s \cos t - \sin s \sin t $$

Determine the signs of $$ \sin(s+t) $$ and $$ \cos(s+t)  to $$identify the quadrant of $$ s + t .$$

Use the given information to find the quadrant of s + t. See Example 3.
cos s = - 15/17 and sin t = 4/5, s in quadrant II and t in quadrant I

Verified video answer for a similar problem:

Key Concepts

Trigonometric Ratios and Signs in Quadrants

Sum of Angles and Quadrant Determination

Using Pythagorean Identity to Find Missing Ratios