Math CentralQuandaries & Queries


Solve the given equation for A, giving your answers in the interval from -180 to 180
4 sin A cos A + 1 = 2 (sin A + cosA)

Hi Carla,

I want to use the expression that cos2(A) = 1 - sin2(A) so my first though is "what happens if I square both sides?" Squaring gives me
16 sin2(A)cos2(A) + 8 sin(A)cos(A) + 1 = 4(sin2(A) + 2 sin(A)cos(A) + cos2(A))
The sin(A) cos(A) terms cancel and now you can substitute cos2(A) = 1 - sin2(A). This gives a quadratic equation in sin2(A) which factors and should allow you to complete the problem.

If you need more assistance write back,


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 cos2 (A) = 1 - sin2(A), I end up with:
16sin2(A) - 16sin4(A) = 3
I am stuck there, could you show me the whole solution?

Well done Carl, that's the equation I got also. Rewrite it as

16 sin4(a) - 16 sin2(A) + 3 = 0

Now let x = sin2(A) and substitute into the equation to get

16 x2 - 16 x + 3 = 0

Can you factor this quadratic and solve for x?



Dear Harley,

Here is my solution


Square both sides of the equation:
16sin2Acos2 + 8sinAcosA + 1 = 4(sin2A + 2sinAcosA+cos2A)
16sin2Acos2 + 1 = 4sin2A + 4cos2A
Substitute cos2A = 1 - sin2A
16sin2A(1 - sin2A) + 1 = 4sin2A + 4(1 - sin2A)
16sin2A - 16sin4A + 1=4sin2A + 4 - 4sin2A
16sin2A - 16sin4A + 1 = 4
Substitute sin2A = X in order to obtain a quadratic equation:
-16X2 + 16X - 3=0
We obtain 2 possible values for sin2A:
sin2A = 3/4 or sin2A = 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...



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.


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