Find two standard position primary angles in radians by solving for the unknown. "Primary angles" are those angles which exist between 0 and 2pi. As usual, use exact values in your calculations.
cosx - 2cos^2x = 0
cosx(1 - 2cosx)=0
cosx=0 or cosx=1/2
x = pi/2, 3pi/2, pi/3, 5pi/3 (90,270,60,300º)
To find the standard position primary angles in radians, we need to solve the given equation:
cosx - 2cos^2x = 0
Step 1: Rewrite the equation using the double-angle identity for cosine:
cosx - 2(1 - sin^2x) = 0
Step 2: Simplify the equation:
cosx - 2 + 2sin^2x = 0
2sin^2x + cosx - 2 = 0
Step 3: Rearrange the equation to isolate sin^2x:
2sin^2x = -cosx + 2
sin^2x = (-cosx + 2)/2
sin^2x = (2 - cosx)/2
Step 4: Take the square root of both sides to solve for sinx:
sinx = ± √((2 - cosx)/2)
Step 5: Now we can solve for the primary angles by using the inverse sine function. Remember that the inverse sine function returns the principal value between -π/2 and +π/2, so we need to find the angles within that range:
For sinx = √((2 - cosx)/2):
x1 = arcsin(√((2 - cosx)/2))
For sinx = -√((2 - cosx)/2):
x2 = -arcsin(√((2 - cosx)/2))
Step 6: Substitute the value of cosx into the equations x1 and x2 to get the final values of the angles.
Note: Since we don't have the exact value of cosx, we cannot determine the exact values of x1 and x2. However, you can input different values for cosx within the range of -1 to 1 and compute the corresponding x values.
Remember to convert the angles from radians to degrees if necessary by using the conversion factor 180/π.