Question

Question #f5498

Answer 1

See below.

Explanation:

I don't know if you are working in degrees or radians, so we will do this in degrees. We need to specify an interval to work in, let's make this `color(blue)(0^o<=theta<=360^o)`

We first need to find what angle corresponds to a tangent of `-1`.
On your calculator there will be a key marked `color(blue)(tan^-1)`. press this key and then enter `-1`. It will return the result `color(blue)(-45^o)`. The reason it returns this value and no others is due to functions and their inverses, and we will not get involved with this now.

The angle `-45^o` is measured from the positive x axis in a clockwise direction, this is equivalent to the positive angle:

`360^o-45^o=315^o`

We know the the tangent ratio is negative in the II quadrant and the IV quadrant. We have the angle in the IV quadrant now we look for an angle in the II quadrant. If we measure an angle of `45^o` from the negative x axis in a clockwise direction, this will be equivalent to an angle:

`180^o-45^o=135^o`

This is our second angle. So the two angles in the given interval are:

`135^o , 315^o`

Answer 2

135^@, or (3pi)/4
315^@, or (7pi)/4

Explanation:

There are 2 ways to solve a trig equation: tan t = -1
a. Use the Trig Table of Special Arcs (Angles) and the Unit Circle.
The Trig Table gives -->
tan t = -1 --> `t = 135^@`, or in radians, `t = (3pi)/4`
The unit circle gives another t that has the same tan value (-1) -->
`t = 135 + 180 = 315^@`, or in radians, `t = (3pi)/4 + pi = (7pi)/4`

b. Use calculator and unit circle.
tan t = - 1
Calculator gives `t = - 45^@`, or `t = 315^@` (co-terminal)
Unit circle gives another `t = - 45 + 180 = 135^@`