AnswersPrecalculusPractice TestQuestion 8
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Question 8 • Precalculus

Based on the diagram, which equation can be simplified to derive the cosine sum identity?

Answer
A
square root of open paren cosine of open paren u plus v close paren minus 1 close paren squared plus open paren sine of open paren u plus v close paren minus 0 close paren squared end root is equal to square root of open paren cosine of negative v minus 1 close paren squared plus open paren sine of negative v minus 0 close paren squared end root
B
square root of open paren cosine of open paren u plus v close paren minus 1 close paren squared plus open paren sine of open paren u plus v close paren minus 0 close paren squared end root is equal to square root of open paren cosine of u minus cosine of negative v close paren squared plus open paren sine of u minus sine of negative v close paren squared end root
C
square root of open paren cosine of open paren u plus v close paren minus 1 close paren squared plus open paren sine of open paren u plus v close paren minus 0 close paren squared end root is equal to square root of open paren cosine of open paren u plus v close paren minus cosine of u close paren squared plus open paren sine of open paren u plus v close paren minus sine of u close paren squared end root
D
square root of open paren cosine of open paren u plus v close paren minus 1 close paren squared plus open paren sine of open paren u plus v close paren minus 0 close paren squared end root is equal to square root of open paren cosine of u minus 1 close paren squared plus open paren sine of u minus 0 close paren squared end root

Explanation

A point at angle u + v on the unit circle is separated from (1, 0) by the same angle as the points at angles u and −v, since u − (−v) = u + v. Thus their distances are equal, which is the equation in choice B. Squaring both sides and using cos(−v) = cos v and sin(−v) = −sin v lets you simplify the equation to obtain the cosine sum identity, cos(u + v) = cos u cos v − sin u sin v.

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