For what values of x is the trigonometric equation tan(x − 180°) − sin(x − 180°) = 0, where 0 ≤ x < 360°, true? {0°, 90°}
{0°, 180°}
{90°, 180°}
{90°, 270°}

It's the second option {0°, 180°}

Explanation:

Given a trigonometric equation in degrees where tan(x − 180°) − sin(x − 180°) = 0 .

Since this is multiple choice, you can use the process of elimination to get rid of the incorrect answers.

You can think of sin as the y or vertical component of a rotation, cos as the x or horizontal component of a rotation, and tangent as we know to be sin/cos. Remember, this rotation is oriented in the counterclockwise direction so it is going leftward starting at (1,0). We also know that 360° is a full rotation so the rotation rule for each interval of 90 degrees counter clockwise is 0° or 360° (x,y) → 90° (-y,x) → 180° (-x,-y) → 270° (y,-x).

So if sin(90°) is 1, cos must be 0 as (1,0) → (0,1) , which means tan is N/A (undefined) since 1/0 = N/A. And undefined + 1 ≠ 0 so anything with 90° is incorrect. That leaves us with the second option {0°, 180°} because it is the only one that excludes 90°.