Question

Which of the following is a solution to sin(2x) - cos(x) = 0?

1) 60
2) 90
3) 120
4) 240
5) 300

Answer 1

To solve sin(2x) - cos(x) = 0, we use the double angle identity and find that x = 90° and x = 270° when cos(x) = 0 and x = 30° for 2sin(x) - 1 = 0. Among the given options, 90° and 300° are solutions.

Step-by-step explanation:

The question asks for a solution to the equation sin(2x) - cos(x) = 0. A key step in solving this type of trigonometric equation is to express all terms in terms of a single trigonometric function if possible. We can use the double angle identity, sin(2x) = 2sin(x)cos(x), to rewrite the equation.

Let's apply the identity to the equation:

1. 2sin(x)cos(x) - cos(x) = 0
2. cos(x)(2sin(x) - 1) = 0

From the second step, we can find solutions by setting each factor equal to zero:

• cos(x) = 0
• 2sin(x) - 1 = 0

For cos(x) = 0, the solutions are x = 90° and x = 270° in the range of 0° to 360°. For 2sin(x) - 1 = 0, the solution is x = 30°.

Comparing these solutions to the options given, we see that option 2) 90 and option 5) 300 (which is an equivalent angle to -60, or 360-60) degrees are correct.