What is the exact value of sine of open paren v plus w close paren , given sine v is equal to negative 3 fifths , cosine w is equal to 10 over 11 , v is an angle in Quadrant III, and w is an angle is Quadrant IV?
Answer
A
the fraction with numerator negative 33 plus 5 square root of 21 and denominator 55
B
the fraction with numerator negative 30 plus 4 square root of 21 and denominator 55
C
the fraction with numerator negative 33 minus 5 square root of 21 and denominator 55
D
the fraction with numerator negative 30 minus 4 square root of 21 and denominator 55
Explanation
Use the sum identity: sin(v + w) = sin v cos w + cos v sin w. Since v is in Quadrant III, cos v is negative, so cos v = −4/5; since w is in Quadrant IV, sin w is negative, so sin w = −√21/11. Substituting gives (−3/5)(10/11) + (−4/5)(−√21/11) = (−30 + 4√21)/55. Thus the exact value is choice D.
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