Trigonometric Identities - Math Is Fun

Trigonometric Identities

You might like to read about Trigonometry first!

Right Triangle

The Trigonometric Identities are equations that are true for right triangles.

(If it isn't a right triangle use the Triangle Identities page)

Each side of a right triangle has a name:

Adjacent is always next to the angle

And Opposite is opposite the angle

We'll soon play with all sorts of functions, but remember it all comes back to that simple triangle with:

• Angle θ
• Hypotenuse
• Adjacent
• Opposite

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 :

Sine Function:sin(θ) = Opposite / Hypotenuse Cosine Function:cos(θ) = Adjacent / Hypotenuse Tangent Function:tan(θ) = Opposite / Adjacent

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

More About Tan

When we divide Sine by Cosine we get:

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

So we can say:

tan(θ) = sin(θ)cos(θ)

That's our first Trigonometric Identity.

Cosecant, Secant and Cotangent

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

Cosecant Function:csc(θ) = Hypotenuse / Opposite Secant Function:sec(θ) = Hypotenuse / Adjacent Cotangent Function:cot(θ) = Adjacent / Opposite

Example: when Opposite = 2 and Hypotenuse = 4 then

sin(θ) = 2/4, and csc(θ) = 4/2

Because of all that we can say:

sin(θ) = 1/csc(θ)

cos(θ) = 1/sec(θ)

tan(θ) = 1/cot(θ)

And the other way around:

csc(θ) = 1/sin(θ)

sec(θ) = 1/cos(θ)

cot(θ) = 1/tan(θ)

And we also have:

cot(θ) = cos(θ)/sin(θ)

Some more Trigonometric Identities for you!

Pythagoras Theorem

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

The Pythagorean Theorem says, in a right triangle, the square of a plus the square of b is equal to 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

• a/c is Opposite / Hypotenuse, which is sin(θ)
• b/c is Adjacent / Hypotenuse, which is cos(θ)

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

sin2 θ + cos2 θ = 1

• sin2 θ means to find the sine of θ, then square the result, same as (sin θ)2
• but sin θ2 means to square θ, then do the sine function

Example: 32°

Using 4 decimal places only:

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

Now let's calculate sin2 θ + cos2 θ:

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

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

Related identities include:

sin2 θ = 1 − cos2 θ cos2 θ = 1 − sin2 θ tan2 θ + 1 = sec2 θ tan2 θ = sec2 θ − 1 cot2 θ + 1 = csc2 θ cot2 θ = csc2 θ − 1

How Do You Remember Them? The identities mentioned so far can be remembered using one clever diagram called the Magic Hexagon:

But Wait ... There's 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

sin 2θ = 2 sin θ cos θ = 2 tan θ1 + tan2 θ cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ = 1 − tan2 θ1 + tan2 θ tan 2θ = 2 tan θ1 − tan2 θ

Half Angle Identities

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

Angle Sum and Difference Identities

Note that ± means you can use plus or minus, and the ∓ means to 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 to all triangles (not just right triangles)

741, 1554, 742, 1555, 743, 1556, 744, 3969, 1557, 3970 Sine, Cosine and Tangent Unit Circle