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:

The length of one side of a right angled triangle,
divided by another side


... but you must know which sides!

triangle showing Opposite, Adjacent and Hypotenuse

For a triangle with 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 you 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)

Quadrants

In Quadrant I both x and y are positive, but ...

  • 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, while 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 they 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:

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.


This graph shows "ASTC" also.

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)