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(52879)   (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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

To facilitate my writing, I will use variable x instead of "  λ ".


Since sec(x) = ,  we have  cos(x) =  = .


Next, sin(x) =  =  =  =  = .


      (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) = ;  sin(x) = ;  cot(x) =  = .


Now  cos^4 (x)-2sin^2 (x) +cot^3 (x) =  = use your calculator = 1.6033    (rounded).    ANSWER

Solved.



Answer by greenestamps(13209)   (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 ; cosine is the reciprocal, .

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 (because ).

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.
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