What is the value of theta if 9 cos^2 theta + 3 cos theta = 0 where 90 degrees < theta < 180 degrees?
To find the value of theta in the equation 9cos^2(theta) + 3cos(theta) = 0, where 90 degrees < theta < 180 degrees, we can solve the equation step by step.
Step 1: Factor out cos(theta) from the equation.
cos(theta)(9cos(theta) + 3) = 0
Step 2: Set each factor equal to zero and solve for theta.
cos(theta) = 0 or 9cos(theta) + 3 = 0
Step 3: Solve for theta when cos(theta) = 0.
A cosine function is equal to zero at 90 degrees and 270 degrees. However, we're considering the range where 90 degrees < theta < 180 degrees. Therefore, the cos(theta) = 0 solutions that satisfy the given range are:
theta = 90 degrees
Step 4: Solve for theta when 9cos(theta) + 3 = 0.
Subtract 3 from both sides of the equation:
9cos(theta) = -3
Divide both sides by 9:
cos(theta) = -1/3
To find theta, we can take the inverse cosine (cos^(-1)) of -1/3. However, the range we're interested in is 90 degrees < theta < 180 degrees, which is in the second quadrant. In the second quadrant, cos(theta) is negative.
Using a calculator, find the inverse cosine of -1/3, which is approximately 109.47 degrees. However, this value is in the first quadrant. In the second quadrant, the angle is equal to 180 degrees minus the angle in the first quadrant.
Therefore, the solution is:
theta = 180 degrees - 109.47 degrees
theta = 70.53 degrees
In conclusion, the value of theta in the given equation where 90 degrees < theta < 180 degrees is 70.53 degrees.