Section 4: sine and cosine rule


You might lượt thích to read about Trigonometry first!

Right Triangle

The Trigonometric Identities are equations that are true for Right Angled Triangles. (If it is not a Right Angled Triangle go to lớn the Triangle Identities page.)

Each side of a right triangle has a name:


Adjacent is always next to lớn the angle

And Opposite is opposite the angle

We are soon going to be playing with all sorts of functions, but remember it all comes back to that simple triangle with:

Angle θHypotenuseAdjacentOpposite

Sine, Cosine and Tangent

The three main functions in trigonometry are Sine, Cosine and Tangent.

They are just the length of one side divided by another

For a right triangle with an angle θ :


tan(θ) = Opposite / Adjacent

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

When we divide Sine by Cosine we get:

sin(θ)cos(θ) = Opposite/HypotenuseAdjacent/Hypotenuse = OppositeAdjacent = tan(θ)

So we can say:

That is our first Trigonometric Identity.

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Cosecant, Secant and Cotangent

We can also divide "the other way around" (such as Adjacent/Opposite instead of Opposite/Adjacent):

Pythagoras Theorem

For the next trigonometric identities we start with Pythagoras" Theorem:


The Pythagorean Theorem says that, in a right triangle, the square of a plus the square of b is equal khổng lồ the square of c:

a2 + b2 = c2

Dividing through by c2 gives

a2c2 + b2c2 = c2c2

This can be simplified to:

(ac)2 + (bc)2 = 1

Now, a/c is Opposite / Hypotenuse, which is sin(θ)

And b/c is Adjacent / Hypotenuse, which is cos(θ)

So (a/c)2 + (b/c)2 = 1 can also be written:

Note:sin2 θ means lớn find the sine of θ, then square the result, andsin θ2 means lớn square θ, then bởi vì the sine function

Example: 32°

Using 4 decimal places only:

sin(32°) = 0.5299...cos(32°) = 0.8480...

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Now let"s calculate sin2 θ + cos2 θ:

0.52992 + 0.84802 = 0.2808... + 0.7191... = 0.9999...

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We get very close lớn 1 using only 4 decimal places. Try it on your calculator, you might get better results!

sin2 θ = 1 − cos2 θcos2 θ = 1 − sin2 θtan2 θ + 1 = sec2 θtan2 θ = sec2 θ − 1cot2 θ + 1 = csc2 θcot2 θ = csc2 θ − 1

How vì chưng You Remember Them?

The identities mentioned so far can beremembered using one clever diagram called the Magic Hexagon:


But Wait ... There is More!

There are many more identities ... Here are some of the more useful ones:

Opposite Angle Identities

sin(−θ) = −sin(θ)

cos(−θ) = cos(θ)

tan(−θ) = −tan(θ)

Double Angle Identities


Half Angle Identities

Note that "±" means it may be either one, depending on the value of θ/2


Angle Sum và Difference Identities

Note that means you can use plus or minus, và the means khổng lồ use the opposite sign.

sin(A B) = sin(A)cos(B) cos(A)sin(B)

cos(A B) = cos(A)cos(B) sin(A)sin(B)

tan(A B) = tan(A) tan(B)1 tan(A)tan(B)

cot(A B) = cot(A)cot(B) 1cot(B) cot(A)

Triangle Identities

There are also Triangle Identities which apply lớn all triangles (not just Right Angled Triangles)