SOLUTION: If tan &#952; = {{{ 3/2 }}} & cos &#952; < 0, use the fundamental identities to evaluate the remaining five trigonometric functions of theta.

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Question 1067569: If tan θ = +3%2F2+ & cos θ < 0, use the fundamental identities to evaluate the remaining five trigonometric functions of theta.
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
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If tan θ = +3%2F2+ & cos θ < 0, use the fundamental identities to evaluate the remaining five trigonometric functions of theta.
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If tan%28theta%29 = 3%2F2 and cos%28theta%29 < 0,  then the angle theta is in QIII.


Next, sin%5E2%28theta%29 = 1%2F%281+%2B+%281%2Ftan%5E2%28theta%29%29%29 is the fundamental identity (which means "Everybody must know it").


Therefore, sin%5E2%28theta%29 = 1+%2F%281+%2B+%281%2F%283%2F2%29%5E2%29%29%29 = 1%2F%281+%2B+%281%2F%28%289%2F4%29%29%29%29 = 1%2F%281%2B4%2F9%29 = 1%2F%28%2813%2F9%29%29 = 9%2F13.

            Hence,  sin%28theta%29 = -sqrt%289%2F13%29 = -3%2Fsqrt%2813%29.    The sign is "-" ("minus") since theta is in QIII.



Further,   cos%28theta%29 = -sqrt%281-sin%5E2%28theta%29%29 = -sqrt%281-9%2F13%29 = -sqrt%284%2F13%29 = -2%2Fsqrt%2813%29.    The sign is "-" ("minus") since theta is in QIII.



Now you know enough to complete the assignment on your own.


The rest is simply Arithmetic plus knowledge of basic definitions of Trigonometry.