Solve the multiple-angle equation.
cos x/2 = - squareroot 3/2
sorry that is supposed to be math not amth!
I know cos 30° or cos π/6 = +√3/2
so x/2 = 180-30 or x/2 = 180+30
x = 300° or x = 420°
or
x = 5π/3 or 7π/3
To do these type of questions, you MUST know the ratio of sides of the 30-60-90° as well as the 45-45-90° triangles, and you MUST be able to recognize the trig ratios.
e.g. I immediately saw that √3/2 came from the 30-60-90 triangle.
To solve the multiple-angle equation cos(x/2) = -√3/2, we can use the properties of the unit circle and the trigonometric functions. Here's how you can go about solving it step by step:
1. Start by recognizing that the value -√3/2 corresponds to the cosine of a special angle. In this case, it corresponds to a 120-degree angle or a 2π/3 angle in radians.
2. Recall that the cosine function is positive in the first and fourth quadrants. So, to find all solutions to the equation, we need to determine the reference angle that corresponds to the given cosine value.
3. Divide the given angle 2π/3 or 120 degrees by 2 to find the reference angle. In this case, the reference angle is π/3 or 60 degrees.
4. Since cosine is positive in the first quadrant, the solutions for x/2 fall within the range of 0 to π/3 (0 to 60 degrees).
5. To find all the solutions, we add an integer multiple of the period to our solution. Since the period of cosine is 2π, we can add multiples of 2π to our solution.
6. Combining these steps, we have the solutions for x/2 as follows: x/2 = π/3 + 2πn, where n is an integer.
7. To find the solutions for x, multiply both sides of the equation by 2: x = 2(π/3 + 2πn).
8. Simplify the expression: x = 2π/3 + 4πn, where n is an integer.
Therefore, the solutions to the multiple-angle equation cos(x/2) = -√3/2 are x = 2π/3 + 4πn, where n is an integer.