SOLUTION: Please help me with this question, Given that tan(x) = a/b, 180° ≤ x ≤ 270°, evaluate cos(4x)

Algebra ->  Trigonometry-basics -> SOLUTION: Please help me with this question, Given that tan(x) = a/b, 180° ≤ x ≤ 270°, evaluate cos(4x)      Log On


   

Question 1190066: Please help me with this question, Given that tan(x) = a/b, 180° ≤ x ≤ 270°, evaluate cos(4x)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Given:
tan%28x%29=a%2Fb
sin%28x%29%2Fcos%28x%29+=+a%2Fb+=>sin%28x%29=a and cos%28x%29+=b

we will have real solution if
b%3C%3E0, a%3E0, b%3E=0 when 180° ≤ x ≤ 270°

use identity:
+cos%284x%29=sin%5E4%28x%29+%2B+cos%5E4%28x%29+-+6+sin%5E2%28x%29+cos%5E2%28x%29

+cos%284x%29=a%5E4+%2B+b%5E4+-+6a%5E2%2A+b%5E2

cos%284x%29=%28a%5E2+-+2ab+-+b%5E2%29+%28a%5E2+%2B+2ab+-+b%5E2%29


Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
Please help me with this question, Given that tan(x) = a/b, 180° ≤ x ≤ 270°, evaluate cos(4x)
~~~~~~~~~~~~~~~~~


            In the post by @MathLover1,  there are mistakes that lead to incorrect answer.

            I came to bring a correct solution. 


Given  tan(x) = a%2Fb.


It implies  sin%5E2%28x%29 = a%5E2%2F%28a%5E2%2Bb%5E2%29,  cos%5E2%28x%29 = b%5E2%2F%28a%5E2%2Bb%5E2%29.


Use identity:    cos(4x) = sin^4(x) + cos^4(x) - 6 sin^2(x) cos^2(x).


It gives


        cos(4x) = a%5E4%2F%28a%5E2%2Bb%5E2%29%5E2 + b%5E4%2F%28a%5E2%2Bb%5E2%29%5E2 - %286a%5E2%2Ab%5E2%29%2F%28a%5E2%2Bb%5E2%29%5E2     (1)

    or  

        cos(4x) = %28a%5E4%2Bb%5E4-6a%5E2%2Ab%5E2%29%2F%28a%5E2%2Bb%5E2%29%5E2.     (2)


Any of formulas (1) or (2) is the ANSWER.