Sine, Cosine, Tangent

Three Functions, but same idea.

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Right Triangle

Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.

Before getting stuck into the functions, it helps lớn give a name lớn each side of a right triangle:

triangle showing Opposite, Adjacent and Hypotenuse

"Opposite" is opposite lớn the angle θ
"Adjacent" is adjacent (next to) lớn the angle θ
"Hypotenuse" is the long one

examples of Opposite, Adjacent and Hypotenuse

Adjacent is always next lớn the angle

And Opposite is opposite the angle

Sine, Cosine and Tangent

Sine, Cosine and Tangent (often shortened lớn sin, cos and tan) are each a ratio of sides of a right angled triangle:

sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent

For a given angle θ each ratio stays the same
no matter how big or small the triangle is

To calculate them:

Divide the length of one side by another side

Example: What is the sine of 35°?

triangle with 2.8, 4.0 and 4.9 sides

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

sin(35°)	= OppositeHypotenuse
	= 2.84.9
	= 0.57...

cos(35°)	= AdjacentHypotenuse
	= 4.04.9
	= 0.82...

tan(35°)	= OppositeAdjacent
	= 2.84.0
	= 0.70...

Size Does Not Matter

The triangle can be large or small and the ratio of sides stays the same.

Only the angle changes the ratio.

Try dragging point "A" lớn change the angle and point "B" lớn change the size:

algebra/images/sin-cos-tan.js

calculator-sin-cos-tan

Good calculators have sin, cos and tan on them, lớn make it easy for you. Just put in the angle and press the button.

But you still need lớn remember what they mean!

In picture form:

sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent individual illustration

Practice Here:

Sohcahtoa

How lớn remember? Think "Sohcahtoa"!

It works lượt thích this:

Soh...

Sine = Opposite / Hypotenuse

...cah...

Cosine = Adjacent / Hypotenuse

...toa

Tangent = Opposite / Adjacent

You can read more about sohcahtoa ... please remember it, it may help in an exam !

Angles From 0° lớn 360°

Move the mouse around lớn see how different angles (in radians or degrees) affect sine, cosine and tangent.

algebra/images/circle-triangle.js

In this animation the hypotenuse is 1, making the Unit Circle.

Xem thêm: văn tả cảnh công viên lớp 5

Notice that the adjacent side and opposite side can be positive or negative, which makes the sine, cosine and tangent change between positive and negative values also.

"Why didn't sin and tan
go lớn the party?"
"... just cos!"

Examples

Example: what are the sine, cosine and tangent of 30° ?

The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √3:

30 degree triangle

Now we know the lengths, we can calculate the functions:

Sine

sin(30°) = 1 / 2 = 0.5

Cosine

cos(30°) = 1.732 / 2 = 0.866...

Tangent

tan(30°) = 1 / 1.732 = 0.577...

(get your calculator out and kiểm tra them!)

Example: what are the sine, cosine and tangent of 45° ?

The classic 45° triangle has two sides of 1 and a hypotenuse of √2:

45 degree triangle

Sine

sin(45°) = 1 / 1.414 = 0.707...

Cosine

cos(45°) = 1 / 1.414 = 0.707...

Tangent

tan(45°) = 1 / 1 = 1

Why?

Why are these functions important?

Because they let us work out angles when we know sides
And they let us work out sides when we know angles

trig example

Example: Use the sine function lớn find "d"

We know:

The cable makes a 39° angle with the seabed
The cable has a 30 meter length.

And we want lớn know "d" (the distance down).

Start with:sin 39° = opposite/hypotenuse

sin 39° = d/30

Swap Sides:d/30 = sin 39°

Use a calculator lớn find sin 39°: d/30 = 0.6293...

Multiply both sides by 30:d = 0.6293… x 30

d = 18.88 lớn 2 decimal places.

The depth "d" is 18.88 m

Exercise

Try this paper-based exercise where you can calculate the sine function for all angles from 0° lớn 360°, and then graph the result. It will help you lớn understand these relatively simple functions.

You can also see Graphs of Sine, Cosine and Tangent.

And play with a spring that makes a sine wave.