Trigonometric Identities

Right Triangle

The Trigonometric Identities are equations that are true for Right Angled Triangles.

Each side of a right triangle has a name:

Trigonometric Identities Right Triangle
  • Adjacent is always next to the angle
  • Opposite is opposite the angle

Sine, Cosine and Tangent

3 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 Functionsin(θ) = Opposite / Hypotenuse
Cosine Functioncos(θ) = Adjacent / Hypotenuse
Tangent Functiontan(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:

Trigonometric Identities Sin Cos Tan

So we can say:

Trigonometric Identities Tan Sin Cos

Cosecant, Secant and Cotangent

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

Cosecant Functioncsc(θ) = Hypotenuse / Opposite
Secant Functionsec(θ) = Hypotenuse / Adjacent
Cotangent Functioncot(θ) = Adjacent / Opposite

Example

when Opposite = 1 and Hypotenuse = 2 then

sin(θ) = 1/2, and

csc(θ) = 2/1

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)

Pythagoras Theorem

Trigonometric Identities Phytagoras

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 to the square of c:

a2 + b2 = c2

Dividing through by c2 gives

Trigonometric Identities 2

This can be simplified to:

Trigonometric Identities
  • 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

Note:

sin2 θ means to find the sine of θ, then square the result, and

sin θ2 means to square θ, then do the sine function

Related identities

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

Opposite Angle Identities

sin(−θ) = −sin(θ)

cos(−θ) = cos(θ)

tan(−θ) = −tan(θ)

Double Angle Identities

Trigonometric Identities Double Angles

Half Angle Identities

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

Trigonometric Identities Half Angles

Angle Sum and Difference Identities

Sinesin (A+B) = sin A cos B + cos A sin B
sin (A-B) = sin A cos B – cos A sin B
Cosinecos (A+B) = cos A cos B – sin A sin B
cos (A-B) = cos A cos B + sin A sin B
TangentTrigonometric Identities Sum Tan
CotangentTrigonometric Identities Sum Cot

Learn More

Solving SSS Triangles

Sohcahtoa

Solving SSA Triangles

Finding an Angle in a Right Angled Triangle

Algebra Index

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