find soultuion in degree, use exact value
1. cos2xcosx+sin2xsinx=-1/2
Do you mean cos (2x) or cos^2 x?
Do you mean sin (2x) or sin^2 x?
Use the idenitites for cos 2x and sin 2x in terms of sin x and cosx. You can end up with an equation that involves just cos x and sin x. If you take it that far, I can help you the rest of the way.
To find the solution of the equation cos(2x)cos(x) + sin(2x)sin(x) = -1/2 in degrees and using exact values, we can use trigonometric identities and solve for x.
Let's simplify the left side of the equation using the identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
The equation becomes:
cos(2x - x) = -1/2
cos(x) = -1/2
Now, we need to find the angles where cos(x) = -1/2. We can refer to the unit circle or use the inverse cosine function (arccos) to find these angles.
On the unit circle, the angle x can be found at the quadrants where the x-coordinate is -1/2. These quadrants are the second (Q2) and third (Q3) quadrants.
In Q2, the reference angle is A = arccos(1/2) = 60 degrees. So, the solutions in Q2 are 180 - A = 180 - 60 = 120 degrees.
In Q3, the reference angle is A = arccos(1/2) = 60 degrees. So, the solutions in Q3 are 180 + A = 180 + 60 = 240 degrees.
Therefore, the solutions to the equation cos(2x)cos(x) + sin(2x)sin(x) = -1/2 in degrees and using exact values are x = 120 degrees and x = 240 degrees.