How many values of ? (0 = ? < 2??? ) are solutions to this equation? tan ( ? ) = - 1

Question

I have no idea how to do this fam.

Answer 1

Hello Brent.

 

We want to find the angle whose tangent is -1.

 

For the tangent to be -1, the sin and cos must be of equal magnitude but of opposite sign.

 

The reference angle whose sin and cos are equal in magnitude is 45^° or π/4.

 

Now recall the following about the unit circle.

 

In the 1st quadrant, sin, cos and tan are all positive.

In the 2nd quadrant, only sin is positive.

In the 3rd quadrant, only tan is positive.

In the 4th quadrant, only cos is positive.

 

This tells me that tan is negative in the 2nd and 4th quadrants.

 

The angles in the 2nd and 4th quadrants with a reference angel of 45° or π/4 are

135° or 3π/4 and 225° or 5π/4