Hi Carla,
I want to use the expression that cos^{2}(A) = 1  sin^{2}(A) so my first though is "what happens if I square both sides?" Squaring gives me
16 sin^{2}(A)cos^{2}(A) + 8 sin(A)cos(A) + 1 = 4(sin^{2}(A) + 2 sin(A)cos(A) + cos^{2}(A))
The sin(A) cos(A) terms cancel and now you can substitute cos^{2}(A) = 1  sin^{2}(A). This gives a quadratic equation in sin^{2}(A) which factors and should allow you to complete the problem.
If you need more assistance write back,
Harley
Dear Harley,
I am still having difficulties solving this problem: 4sin(A)cos(A) = 2 (sin(A)+cos(A))
After squaring the whole expression and substituting cos^{2} (A) = 1  sin^{2}(A), I end up with:
16sin^{2}(A)  16sin^{4}(A) = 3
I am stuck there, could you show me the whole solution?
Thanks.
Carla
Well done Carl, that's the equation I got also. Rewrite it as
16 sin^{4}(a)  16 sin^{2}(A) + 3 = 0
Now let x = sin^{2}(A) and substitute into the equation to get
16 x^{2}  16 x + 3 = 0
Can you factor this quadratic and solve for x?
Harley
Dear Harley,
Here is my solution
4sinAcosA+1=2(sinA+cosA)
Square both sides of the equation:
16sin^{2}Acos^{2} + 8sinAcosA + 1 = 4(sin^{2}A + 2sinAcosA+cos^{2}A)
16sin^{2}Acos^{2} + 1 = 4sin^{2}A + 4cos^{2}A
Substitute cos^{2}A = 1  sin^{2}A
16sin^{2}A(1  sin^{2}A) + 1 = 4sin^{2}A + 4(1  sin^{2}A)
16sin^{2}A  16sin^{4}A + 1=4sin^{2}A + 4  4sin^{2}A
16sin^{2}A  16sin^{4}A + 1 = 4
Substitute sin^{2}A = X in order to obtain a quadratic equation:
16X^{2 }+ 16X  3=0
We obtain 2 possible values for sin^{2}A:
sin^{2}A = 3/4 or sin^{2}A = 1/4
sinA = 0.866 or sinA = 0.5
There are 2 possible values for angle A:
60deg and 30deg
Possible further solutions to the equation can be obtained using the following formulas:
sinA = sin (180  A)
cos A= cos (A)
therefore, solutions are 60deg, 30deg, +30deg, +60deg, +120deg, +150deg
Substituting each solution in the original equation shows that 30deg and 120deg are not possible (one side of the equation being positive and the other negative)
therefore solutions to the equation in the range 180deg to +180deg are:
60deg, +30deg, +60deg, +150deg.
I was just wondering if there is a neater way of sifting out the solutions as I have had to substitute each solution in the equation to see if it works...
Cheers
Carla
You really do have to check each of the possible solutions you found to see which are actually solutions to the original problem. I don't know any way to check other than what you did.
Harley
