Trig Help please. I need to find the quadrant in which (theta) lies from the info given...?

Question

like this problem
sin (theta) < 0 and cos (theta) < 0
I have looked in my book, other trig books (5), and all over the internet. Can someone tell me how to do this?

Answer 1

You need to determine when the sine and the cosine of a given angle are both negative.

Looking at your unit circle, you can see that the sine and cosine are both positive for all angles between 0 and 90 degrees.

The sine of an angle is positive when the angle is between 0 and 180 degrees. The sine is negative when the angle is between 180 and 360 (or 0) degrees.

The cosine of an angle is positive when the angle is between -90 and 90 degrees. The cosine is negative when the angle is between 90 degrees and 270 (or -90) degrees.

So, for the sine and cosine of theta to both be negative, theta must be both between 180 and 360 AND between 90 and 270.

Thus, theta must be between 180 and 270 degrees, which is the third quadrant.

Answer 2

Okay, So first open these links to see an image of what I am talking about.

http://www.mathdaily.com/lessons/Image:Sine_Cosine_Graph.png

http://www.mathdaily.com/lessons/Image:UnitCircle.png

So, first look at the circle with all of the numbers around it.

The top right section of the circle is the 1st quadrant. (0 to Pi/2)
The top left section of the circle is the 2nd quadrant. (Pi/2 to Pi)
The bottom left section is the 3rd quadrant. (Pi to 3Pi/2)
and thus the bottom right section is the 4th quadrant. (3Pi/2 to 0)

So your answer will be one of these quadrants.

Next, take a look at the picture with the waves.

A sine wave oscillates between -1 and 1.
A cosine wave also oscillates between -1 and 1.

However they have different starting points so at different times they are different points.

So at "0", sine = "0" and cosine = "1" as you can see on the wave.

At "pi/2", sine = "1" and cosine = "0".

So in your question you want to find when sine and cosine are both less than "0".

If we look on the wave picture, both sine and cosine are below zero from "pi" to "3pi/2".

Haha, so finally, we can look back at our circle picture and find the quadrant that is from "pi" to "3pi/2", which is... the Third Quadrant.


Also, if you just look at the Circle Graph, you will see at the top, bottom, left, and right of the graph there are coordinates [(1,0);(0,1); (-1,0); (0,-1). The first number is cosine at any given time theta and the the second number is sine.

Hope this wasn't too complicated of an explanation and good luck. :)

Answer 3

Third Quadrant

Answer 4

sinθ<0 is when sinθ is negative to 0, which would be in quadrant 3 or 4.

cosθ<0 is when cosθ is negative to 0, which would be in quadrant 2 or 3.