SOLUTION: If λ is an angle in the second quadrant and sec λ = - √85/6. Find the value for cos^4 λ-2sin^2 λ +cot^3 λ, as a decimal.

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Question 1186337: If λ is an angle in the second quadrant and sec λ = - √85/6. Find the value for cos^4 λ-2sin^2 λ +cot^3 λ, as a decimal.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52878) About Me  (Show Source):
You can put this solution on YOUR website!
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If λ is an angle in the second quadrant and sec λ = - √85/6.
Find the value for cos^4 λ-2sin^2 λ +cot^3 λ, as a decimal.
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To facilitate my writing, I will use variable x instead of "  λ ".


Since sec(x) = -sqrt%2885%29%2F6,  we have  cos(x) = 1%2Fsec%28x%29 = -6%2Fsqrt%2885%29.


Next, sin(x) = sqrt%281+-+cos%5E2%28x%29%29 = sqrt%281+-+%28-6%2Fsart%2885%29%29%5E2%29 = sqrt%281+-+36%2F85%29 = sqrt%2849%2F85%29 = 7%2Fsqrt%2885%29.


      (in my derivation, I used the fact that the angle x is in the second quadrant, taking the sign "+" at the square root).


So, cos(x) = -6%2Fsqrt%2885%29;  sin(x) = 7%2Fsqrt%2885%29;  cot(x) = cos%28x%29%2Fsin%28x%29 = -6%2F7.


Now  cos^4 (x)-2sin^2 (x) +cot^3 (x) = %28-6%2Fsqrt%2885%29%29%5E4+-+2%2A%287%2Fsqrt%2885%29%29%5E2+%2B+%28-6%2F7%29%5E3 = use your calculator = 1.6033    (rounded).    ANSWER

Solved.



Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


This is basic application of the definition of the 6 trig functions, plus some simple arithmetic....

The secant is -%28sqrt%2885%29%2F6%29; cosine is the reciprocal, -%286%2Fsqrt%2885%29%29.

The Pythagorean Theorem, along with the definitions of sine and cosine for a right triangle, and with the angle in quadrant 2, gives us a sine of 7%2Fsqrt%2885%29 (because %28-6%29%5E2%2B%287%29%5E2=85).

Then use the definition of cotangent to find the cotangent of the angle.

The actual computations need to be done on a calculator, which you can do as easily as we can.