Question

Find all angles between 0 and 2pi satisfying the condition cosx=-1/2

Answer 1

The angles satisfying cosx = -1/2 between 0 and 2pi are 2π/3 and 4π/3.

Step-by-step explanation:

To find all the angles between 0 and 2π that satisfy the condition cosx = -1/2, we need to recall that the cosine function is negative in the second and third quadrants.

The reference angle for which the cosine is 1/2 is pi/3 or 60 degrees.

Therefore, the angles we are looking for in the domain [0, 2pi) are pi - pi/3 (2nd quadrant) and pi + pi/3 (3rd quadrant).

The answers are: x = 2π/3 and x = 4π/3.

Answer 2

You can find the angles using the unit circle. You can also find the angle using algebra.

I want to show you the algebra way.

Since cos(x) = cos(2•pi - x), this leads to two equations.

The two equations are as follows:

cos (x) = -1/2...the given problem.

We also have cos (2•pi - x) = -1/2.

To isolate x for both trigonometric equations, use the inverse trig function idea.

Equation 1

arccos(cos x) = arccos (-1/2)

x = 2•pi/3

Equation 2

arccos(cos (2•pi - x)) = arccos (-1/2).

Note: 2•pi - x = 2•pi/3. We are not done with this equation. We must isolate x. In other words, like any other equation, solving for x is needed.

When we subtract 2•pi from both sides, the answer for x in the second equation is 4•pi/3.

As you know, 2•pi/3 and 4•pi/3 lie between 0 and 2•pi on the unit circle.

Answer: 2•pi/3, 4•pi/3