# 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 = |

## 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 counter-clockwise 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 | (−5,4) |

III |
Negative |
Negative |
(−2,−1) |

IV |
Positive | Negative |
(4,−3) |

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

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:

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°

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 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
**A**dd**S**ugar**T**o**C**offee

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?

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)