cos(theta)=-1/2 List 6 specific solutions.

Answer 1

Check out your unit circle. (Google "unit circle" if you don't have one, and get in touch to video chat if you don't understand where it comes from.) The (x,y) coordinates at each angle are the cosine and sine of that angle.

You'll see that at 120^o, or 2π/3 and at 240^o, or 4π/3, the x-coordinate is -1/2. Those are the two angles you start with. In degrees, add or subtract multiples of 360^o from either one to get another answer, as that will take you around the circle another time/s. For example, since 120^o works, so will (120+360)=480^o, (120+360+360) = 840^o, and (120-360)= -240^o.

You can do the same with radians: Instead of adding or subtracting multiples of 360^o, add or subtract multiples of 2π. Since the original angles are measured in thirds of pi, convert 2π to a fraction with 3 in the denominator: 2π = 6π/3. Here are three solutions in addition to 4π/3: (4π/3 - 6π/3)=-2π/3, (4π/3 -6π/3 - 6π/3)= -8π/3, and (4π/3 +6π/3)= 10π/3. You can add or subtract 2π as many times as you want from either 4π/3 or 2π/3 to get infinitely more answers.

I hope that helps! If you're still confused, or if you're not super sure why or how the unit circle works, PLEASE get in touch! It's really fundamental to everything in trigonometry!

Happy studies!

Liz

Answer 2

To determine what angle is associated with a cosine of -1/2, look at the unit circle. Remember that cosine is associated with the x value. The cosine is -1/2 when the angle is 120° and at 240° (or in radian measure that is (2π)/3 and (4π)/3). Those are the only values going around the unit circle one time, when the cosine is -1/2. Since your question is looking for 6 answers, that means you must go around the circle more times.

Add 360° or 2π radians to each answer and you get 480° and 600° (or 8π/3 and 10π/3).

Add another 360° or 2π radians to each of these answers and you get 840° and 960° (or 14π/3 and 16π/3).

Note that there are lots of correct answers. I could have subtracted 360° or 2π radians to get valid answers as well (or could have added or subtracted any multiple of 360° or 2π radians).

Answer 3

Please see video