The facility is put on a graph where the x axis and also y axis cross, so we obtain this neat arrangement here.

You are watching: Cos-1(1/2)


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Sine, Cosine and also Tangent

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


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

cos 0° = 1, sin 0° = 0 and 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! Move the computer mouse around to watch just how various angles (in radians or degrees) impact sine, cosine and tangent


The "sides" have the right to be positive or negative according to the rules of Cartesian coordinates. This renders the sine, cosine and also tangent adjust in between positive and negative values likewise.

Also attempt the Interenergetic Unit Circle.


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Pythagoras

Pythagoras" Theorem states that for a ideal angled triangle, the square of the lengthy side equates to the amount of the squares of the various other two sides:

x2 + y2 = 12

But 12 is just 1, so:

x2 + y2 = 1 equation of the unit circle

Also, given that x=cos and y=sin, we get:

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

Important Angles: 30°, 45° and 60°

You must try to remember sin, cos and tan for the angles 30°, 45° and also 60°.

Yes, yes, it is a pain to have to remember points, but it will certainly make life simpler once you recognize them, not simply in exams, however other times once you have to perform quick estimates, and so on.

These are the worths you must remember!


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

How To Remember?

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To assist you remember, sin goes "1,2,3" :

sin(30°) = 12 = 12 (bereason √1 = 1)

sin(45°) = 22

sin(60°) = 32

And cos goes "3,2,1"

cos(30°) = 32

cos(45°) = 22

cos(60°) = 12 = 12

Just 3 Numbers

In reality, learning 3 numbers is enough:12, √22 and √32

Due to the fact that they occupational for both cos and also sin:

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What around tan?

Well, tan = sin/cos, so we can calculate it prefer 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: creating 1√3 might cost you marks (view Rational Denominators), so instead usage √33

Quick Sketch

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

Draw atriangle via side lengths of 2
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Cut in fifty percent. Pythagoras claims the new side is √3


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

Sine: sohcahtoa


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

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The Whole Circle

For the whole circle we need worths in eextremely quadrant, with the correct plus or minus sign as per Cartesian Coordinates:

Note that cos is first and sin is second, so it goes (cos, sin):


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Example: What is cos(330°) ?

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Make a sketch like this, and we can check out it is the "long" value: √32


And this is the very 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 make a sketch.

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We deserve to then watch it is negative and also is the "short" value: −½


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

Footnote: wright here do the worths come from?

We have the right to usage the equation x2 + y2 = 1 to find the lengths of x and also y (which are equal to cos and also sin once the radius is 1):

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45 Degrees

For 45 degrees, x and y are 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 are 60°) and also separation it down the middle.

The "x" side is now ½,

And the "y" side is:


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

30 Degrees

30° is simply 60° through x and y swapped, so x = √(¾) and also y = ½

And:


√1/2 = √2/4 = √2√4 = √22

Also:


√3/4 = √3√4 = √32

And here is the result (very same as before):


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

Circle Interenergetic Unit Circle Sine, Cosine and Tangent in Four Quadrants Trigonometry Index