Sine, Cosine and Tangent in Four Quadrants

Sine, Cosine and Tangent

The three main functions in trigonometry are Sine, Cosine and Tangent.

triangle showing Opposite, Adjacent and Hypotenuse

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°?

triangle 2.8 4.0 4.9

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:

graph with point (12,5)
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 counter-clockwise direction)

Like this:

Quadrant Signs

Quadrant X
I Positive Positive (3,2)
II Negative Positive  (−5,4)
III Negative Negative (−2,−1)
IV Positive Negative  (4,−3)

cartesian coordinates

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"

Reference Angle

Angles can be more than 90º

But we can bring them back below 90º using the x-axis as the reference.

Think "reference" means "refer x"

The simplest method is to do a sketch!

Example: 160º

Start at the positive x axis and rotate 160º

triangle quadrant example
Then find the angle to the nearest part of the x-axis,
in this case 20º

The reference angle for 160º is 20º

Here we see four examples with a reference angle of 30º:

30 degree reference angles

Instead of a sketch you can use these rules:

Quadrant Reference Angle
I θ
II 180º − θ
III θ − 180º
IV 360º − θ

Sine, Cosine and Tangent in the Four Quadrants

Now let us look at the details of a 30° right 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°

triangle 30 quadrant I

sin(30°) = 1 / 2 = 0.5
cos(30°) = 1.732 / 2 = 0.866
tan(30°) = 1 / 1.732 = 0.577


But in Quadrant II, the x direction is negative, and cosine and tangent become negative:

Example: The sine, cosine and tangent of 150°

triangle 30 quadrant I

sin(150°) = 1 / 2 = 0.5
cos(150°) = −1.732 / 2 = −0.866
tan(150°) = 1 / −1.732 = −0.577


In Quadrant III, sine and cosine are negative:

Example: The sine, cosine and tangent of 210°

triangle 30 quadrant I

sin(210°) = −1 / 2 = −0.5
cos(210°) = −1.732 / 2 = −0.866
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°

triangle 30 quadrant I

sin(330°) = −1 / 2 = −0.5
cos(330°) = 1.732 / 2 = 0.866
tan(330°) = −1 / 1.732 = −0.577

There is a pattern! Look at when Sine Cosine and Tangent are positive ...

This can be shown even easier by:

trig ASTC is All,Sine,Tangent,Cosine

trig graph 4 quadrants
This graph shows "ASTC" also.

Some people like to remember the four letters ASTC by one of these:

Maybe you could make up one of your own. Or just remember ASTC.

Inverse Sin, Cos and Tan

What is the Inverse Sine of 0.5? 

sin-1(0.5) = ?

In other words, when y is 0.5 on the graph below, what is the angle?

sine crosses 0.5 at 30,150,390, etc
There are many angles where y=0.5

The trouble is: a calculator will only give you one of those values ...

... but there are always two values between 0º and 360º
(and infinitely many beyond):

First value Second value
Sine θ 180º − θ
Cosine θ 360º − θ
Tangent θ θ + 180º

We can now solve equations for any angle!

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 next solution is 180º − 30º = 150º (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)

We may need to bring our angle between 0º and 360º by adding or subtracting 360º

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 −52.4º + 180º  = 127.6º (Quadrant II)


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