How do I dertermine the measure of theta for the equation 4cos theta - sqrt 12 = 0. My answer was 2pi/3,4pi/3 I'm told it's wrong could you please show me how to do this type of question
4cos theta - √12 = 0
cos theta = 2√3/4 = √3/2
You should know by memory the ratio of sides of the 30-60-90 triangle
namely that cos 30º = √3/2
so theta is 30º or pi/6 radians
but the cosine is positive in quadrants I and IV.
We have the quadrant I angle, for the IVth one, take 360 - 30 = 300º or
2p - pi/6 = 11pi/6
You also should have memorized the 45-45-90 triangle which is √2 : √2 : 2,
and of course you must know the CAST rule to do these types of questions.
To determine the measure of theta for the equation 4cos(theta) - sqrt(12) = 0, you need to solve for theta.
Step 1: Start by isolating the cosine term. Add sqrt(12) to both sides of the equation:
4cos(theta) = sqrt(12)
Step 2: Divide both sides of the equation by 4:
cos(theta) = sqrt(12)/4
Step 3: Simplify the right side of the equation:
cos(theta) = sqrt(3)/2
Now, to find the values of theta, you need to identify the angles whose cosine is equal to sqrt(3)/2. One way to do this is by using the unit circle or by referring to the special angles in trigonometry.
The cosine of an angle is equal to sqrt(3)/2 in the first and second quadrants. In the unit circle, the angles associated with this cosine value are π/6, 5π/6, 7π/6, and 11π/6.
However, keep in mind that the given equation is 4cos(theta) - sqrt(12) = 0, so we only want the angles where cos(theta) satisfies this equation.
Step 4: Substitute the values of theta back into the given equation to check which angles are valid solutions:
For π/6:
4cos(π/6) - sqrt(12) = 4(sqrt(3)/2) - √12 = 2√3 - √12 ≠ 0
For 5π/6:
4cos(5π/6) - sqrt(12) = 4(-sqrt(3)/2) - √12 = -2√3 - √12 ≠ 0
For 7π/6:
4cos(7π/6) - sqrt(12) = 4(-sqrt(3)/2) - √12 = -2√3 - √12 = 0
For 11π/6:
4cos(11π/6) - sqrt(12) = 4(sqrt(3)/2) - √12 = 2√3 - √12 = 0
Step 5: From the analysis in Step 4, we can see that only the angles 7π/6 and 11π/6 satisfy the given equation. Thus, the values of theta are 7π/6 and 11π/6.
Therefore, the correct answer is theta = 7π/6, 11π/6.