The center is put on a graph where the x axis and y axis cross, for this reason we get this neat arrangement here.

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Sine, Cosine and Tangent

Because the radius is 1, we can straight measure sine, cosine and also tangent.


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What happens when the angle, θ, is 0°?

cos 0° = 1, sin 0° = 0 and also tan 0° = 0


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What happens as soon as θ is 90°?

cos 90° = 0, sin 90° = 1 and also tan 90° is undefined


TryItYourself!

Have a try! relocate the mouse roughly to see how different angles (in radians or degrees) impact sine, cosine and also tangent


The "sides" have the right to be hopeful or an adverse according come the rules of Cartesian coordinates. This provides the sine, cosine and also tangent readjust between positive and an unfavorable values also.

Also shot the interactive Unit Circle.


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Pythagoras

Pythagoras" Theorem states that because that a appropriate angled triangle, the square the the lengthy side amounts to the amount of the squares the the various other two sides:

x2 + y2 = 12

But 12 is just 1, so:

x2 + y2 = 1 equation of the unit circle

Also, because x=cos and also y=sin, us get:

(cos(θ))2 + (sin(θ))2 = 1a valuable "identity"

Important Angles: 30°, 45° and also 60°

You should shot to remember sin, cos and tan because that the angle 30°, 45° and also 60°.

Yes, yes, that is a ache to need to remember things, but it will make life easier when you recognize them, not just in exams, yet other times once you have to do quick estimates, etc.

These room the values you should remember!


angle Sin Cos Tan=Sin/Cos 30° 45° 60°
12 √32 1 √3 = √3 3
√22 √22 1
√32 12 √3

How come Remember?

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To help you remember, sin walk "1,2,3" :

sin(30°) = 12 = 12 (because √1 = 1)

sin(45°) = 22

sin(60°) = 32

And cos go "3,2,1"

cos(30°) = 32

cos(45°) = 22

cos(60°) = 12 = 12

Just 3 Numbers

In fact, learning 3 number is enough:12, √22 and √32

Because they job-related for both cos and sin:

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What about tan?

Well, tan = sin/cos, for this reason we can calculate it like this:

tan(30°) =sin(30°)cos(30°)=1/2√3/2 = 1√3 = √33 *

tan(45°) =sin(45°)cos(45°)=√2/2√2/2 = 1

tan(60°) =sin(60°)cos(60°)=√3/21/2 = √3

* Note: writing 1√3 may cost you marks (see rational Denominators), so instead use √33

Quick Sketch

Another means to help you remember 30° and 60° is to make a quick sketch:

Draw atriangle v side lengths of 2
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Cut in half. Pythagoras claims the brand-new side is √3


12 + (√3)2 = 22
1 + 3 = 4
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Then use sohcahtoa because that sin, cos or tan
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Example: sin(30°)

Sine: sohcahtoa


sine is opposite divided by hypotenuse
sin(30°) = opposite hypotenuse = 1 2

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The whole Circle

For the entirety circle we need values in every quadrant, through the correct plus or minus authorize as per Cartesian Coordinates:

Note that cos is an initial and sin is second, so the goes (cos, sin):


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Save together PDF


Example: What is cos(330°) ?

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Make a sketch choose this, and also we have the right to see that is the "long" value: √32


And this is the same Unit circle in radians.

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Example: What is sin(7π/6) ?

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Think "7π/6 = π + π/6", then do a sketch.

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We have the right to then view it is negative and also is the "short" value: −½


7708, 7709, 7710, 7711, 8903, 8904, 8906, 8907, 8905, 8908

Footnote: where do the values come from?

We have the right to use the equation x2 + y2 = 1 to uncover the lengths the x and y (which room equal come cos and also sin as soon as the radius is 1):

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45 Degrees

For 45 degrees, x and also y room equal, so y=x:


x2 + x2 = 1
2x2 = 1
x2 = ½
x = y = √(½)

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60 Degrees

Take an equilateral triangle (all sides are equal and all angles room 60°) and split it under the middle.

The "x" next is currently ½,

And the "y" next is:


(½)2 + y2 = 1
¼ + y2 = 1
y2 = 1-¼ = ¾
y = √(¾)

30 Degrees

30° is just 60° v x and y swapped, so x = √(¾) and y = ½

And:


√1/2 = √2/4 = √2√4 = √22

Also:


√3/4 = √3√4 = √32

And right here is the an outcome (same together before):


angle Sin Cos Tan=Sin/Cos 30° 45° 60°
12 √32 1 √3 = √3 3
√22 √22 1
√32 12 √3

one interaction Unit one Sine, Cosine and also Tangent in four Quadrants Trigonometry Index