What are the values of θ, between 0 and 2pi, when tan θ=-1 ?

Question

I have already got a response to this question that needs some clearing up. Assuming I don’t want to use the unit circle and just want to type this on my calculator, can someone explain the process (Rule that tangent repeats every 180 explanation would also be appreciated) Thank you for reading

Answer Choices:

F: pi/4 and 3pi/4 only

G: 3pi/4 and 5pi/4 only

H: 3pi/4 and 7pi/4 only

J: 5pi/4 and 7pi/4 only

K: pi/4, 3pi/4, 5pi/4, and 7pi/4 only

Answer 1

Draw a picture

For any line drawn from the origin (0,0), the tangent of the angle is the y coordinate divided by the x coordinate. If the tangent is negative, it means one of these coordinates is negative, but not both. That means the line is in either the 2nd or 4th quadrant.

A tangent of -1 occurs when the angle is pi/4 (45 degrees), plus any multiple of pi (N*pi or n*180)

Angles with a tangent of -1:

135 deg. (3pi/4)

315 deg (7pi/4)

495 deg (11pi/4)

etc.....

Try each one with your calculator

Answer 2

The correct answer is H: 135 and 315 degrees = 3pi/4 and 7pi/4.

I recommend that your calculator be in the "degree mode."

Type the inverse tangent of -1: tan^-1(-1).

The calculator will give you an answer of -45 degrees, which is a 45 degree reference angle in Quadrant IV: 315 degrees = 7pi/4 radians.

The tangent function is also negative in the Quadrant II -- same 45 degree reference angle: 135 degrees = 3pi/4 radians.