Sine, Cosine and Tangent in Four Quadrants
Sine, Cosine and Tangent
The three main functions in trigonometry are Sine, Cosine and Tangent.
They are easy to calculate:
Divide the length of one side of a
right angled triangle by another side
... but we must know which sides!
For an angle θ, the functions are calculated this way:
Sine Function: 
sin(θ) = Opposite / Hypotenuse 
Cosine Function: 
cos(θ) = Adjacent / Hypotenuse 
Tangent Function: 
tan(θ) = Opposite / Adjacent 
Example: What is the sine of 35°?
Using this triangle (lengths are only to one decimal place): sin(35°) = Opposite / Hypotenuse = 2.8/4.9 = 0.57... 
Cartesian Coordinates
Using Cartesian Coordinates we mark a point on a graph by how far along and how far up it is:
The point (12,5) is 12 units along, and 5 units up.
Four Quadrants
When we include negative values, the x and y axes divide the space up into 4 pieces:
Quadrants I, II, III and IV
(They are numbered in a counterclockwise direction)
 In Quadrant I both x and y are positive,
 in Quadrant II x is negative (y is still positive),
 in Quadrant III both x and y are negative, and
 in Quadrant IV x is positive again, and y is negative.
Like this:
Quadrant  X (horizontal) 
Y (vertical) 
Example 

I  Positive  Positive  (3,2) 
II  Negative  Positive  
III  Negative  Negative  (−2,−1) 
IV  Positive  Negative 
Example: The point "C" (−2,−1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction).
Both x and y are negative, so that point is in "Quadrant III"
Sine, Cosine and Tangent in the Four Quadrants
Now let us look at what happens when we place a 30° triangle in each of the 4 Quadrants.
In Quadrant I everything is normal, and Sine, Cosine and Tangent are all positive:
Example: The sine, cosine and tangent of 30°
Sine 
sin(30°) = 1 / 2 = 0.5 
Cosine 
cos(30°) = 1.732 / 2 = 0.866 
Tangent 
tan(30°) = 1 / 1.732 = 0.577 
But in Quadrant II, the x direction is negative, and both cosine and tangent become negative:
Example: The sine, cosine and tangent of 150°
Sine 
sin(150°) = 1 / 2 = 0.5 
Cosine 
cos(150°) = −1.732 / 2 = −0.866 
Tangent 
tan(150°) = 1 / −1.732 = −0.577 
In Quadrant III, sine and cosine are negative:
Example: The sine, cosine and tangent of 210°
Sine 
sin(210°) = −1 / 2 = −0.5 
Cosine 
cos(210°) = −1.732 / 2 = −0.866 
Tangent 
tan(210°) = −1 / −1.732 = 0.577 
Note: Tangent is positive because dividing a negative by a negative gives a positive.
In Quadrant IV, sine and tangent are negative:
Example: The sine, cosine and tangent of 330°
Sine 
sin(330°) = −1 / 2 = −0.5 
Cosine 
cos(330°) = 1.732 / 2 = 0.866 
Tangent 
tan(330°) = −1 / 1.732 = −0.577 
There is a pattern! Look at when Sine Cosine and Tangent are positive ...
 All three of them are positive in Quadrant I
 Sine only is positive in Quadrant II
 Tangent only is positive in Quadrant III
 Cosine only is positive in Quadrant IV
This can be shown even easier by:
This graph shows "ASTC" also.
Some people like to remember the four letters ASTC by one of these:
 All Students Take Chemistry
 All Students Take Calculus
 All Silly Tom Cats
 All Stations To Central
 Add Sugar To Coffee
You can remember one of these, or maybe you could make up
your own. Or
just remember ASTC.

Two Values
Have a look at this graph of the Sine Function:
There are two angles (within the first 360°) that have the same value!
And this is also true for Cosine and Tangent.
The trouble is: Your calculator will only give you one of those values ...
... but you can use these rules to find the other value:
First value  Second value  
Sine  θ  180º − θ 
Cosine  θ  360º − θ 
Tangent  θ  θ − 180º 
And if any angle is less than 0º, then add 360º.
We can now solve equations for angles between 0º and 360º (using Inverse Sine Cosine and Tangent)
Example: Solve sin θ = 0.5
We get the first solution from the calculator = sin^{1}(0.5) = 30º (it is in Quadrant I)
The other solution is 180º − 30º = 150º (Quadrant II)
Example: Solve tan θ = −1.3
We get the first solution from the calculator = tan^{1}(−1.3) = −52.4º
This is less than 0º, so we add 360º: −52.4º + 360º = 307.6º (Quadrant IV)
The other solution is 307.6º − 180º = 127.6º (Quadrant II)
Example: Solve cos θ = −0.85
We get the first solution from the calculator = cos^{1}(−0.85) = 148.2º (Quadrant II)
The other solution is 360º − 148.2º = 211.8º (Quadrant III)