Sine, Cosine and also Tangent

The 3 main functions in trigonometry room Sine, Cosine and also Tangent.

You are watching: Tangent is positive in quadrants i and iv only


They are simple to calculate:

Divide the length of one side of aright angled triangle by another side

... But we must understand which sides!

Using this triangle (lengths are only to one decimal place):

sin(35°) = the contrary / Hypotenuse = 2.8/4.9 = 0.57...

Cartesian Coordinates

Using Cartesian collaborates we note a point on a graph by how far along and also how much up it is:

The point (12,5) is 12 devices along, and also 5 units up.


Four Quadrants

When we incorporate negative values, the x and also y axes division the an are up right into 4 pieces:

Quadrants I, II, III and IV

(They space numbered in a counter-clockwise direction)

In Quadrant I both x and also y room positive, in Quadrant II x is negative (y is tho positive), in Quadrant III both x and also y room negative, andin Quadrant IV x is confident again, and y is negative.

Like this:



Example: The suggest "C" (−2,−1) is 2 units along in the negativedirection, and also 1 unit down (i.e. Negative direction).

Both x and y space negative, therefore that suggest is in "Quadrant III"

Sine, Cosine and also Tangent in theFour Quadrants

Now let us look in ~ what happens when we place a 30° triangle in every of the 4 Quadrants.

In Quadrant I everything is normal, and Sine, Cosine and also Tangent room all positive:

Example: The sine, cosine and tangent of 30°


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 also both cosine and tangent end up being negative:

Example: The sine, cosine and also tangent the 150°


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 also cosine are negative:

Example: The sine, cosine and also tangent of 210°


sin(210°) = −1 / 2 = −0.5
cos(210°) = −1.732 / 2 = −0.866
tan(210°) = −1 / −1.732 = 0.577

Note: Tangent is positive since dividing a an adverse by a an unfavorable gives a positive.

In Quadrant IV, sine and also tangent space negative:

Example: The sine, cosine and also tangent of 330°


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

There is a pattern! watch at once Sine Cosine and also Tangent space positive ...

All 3 of them space positivein Quadrant ISine only is positive in Quadrant IITangent just is positive in Quadrant IIICosine only is optimistic in Quadrant IV

This have the right to be shown even easier by:


Some civilization like to remember the 4 letters ASTC by one of these:

All Students take ChemistryAll Students take CalculusAll silly Tom CatsAll Stations come CentralAdd Sugar To Coffee

You can remember among these, or maybe you can make upyour own. Orjust remember ASTC.

This graph shows "ASTC" also.

2 Values

Have a look in ~ this graph that the Sine Function::

There space two angles (within the an initial 360°) that have the same value!

And this is also true because that Cosine and Tangent.

The problem is: Your calculator will certainly only provide you one of those values ...

See more: Weight Of Wet Concrete Vs Dry Concrete Weigh? Wet Vs Dry Wet Vs Dry

... Yet you deserve to use this rules to find the other value:

First valueSecond value
Sineθ180º − θ
Cosineθ360º − θ
Tangentθθ − 180º

And if any kind of angle is much less than 0º, then add 360º.

We can now settle equations forangles between 0º and also 360º(using train station Sine Cosine and Tangent)

Example: deal with sin θ = 0.5

We get the an initial solution native the calculator = sin-1(0.5) = 30º(it is in Quadrant I)

The other solution is 180º − 30º = 150º (Quadrant II)

Example: Solvetan θ= −1.3

us getthe an initial solution indigenous the calculator = tan-1(−1.3) = −52.4º

This is less than 0º, therefore we add 360º: −52.4º + 360º = 307.6º (Quadrant IV)

The othersolution is307.6º − 180º =127.6º (Quadrant II)

Example: Solvecos θ= −0.85

we getthe an initial solution native the calculator = cos-1(−0.85) =148.2º (Quadrant II)

The other solution is 360º −148.2º = 211.8º (Quadrant III)