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

Angle**θ**HypotenuseAdjacentOpposite

## 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/Hypotenuse***Adjacent/Hypotenuse** = *Opposite***Adjacent** = 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 TheoremFor the next trigonometric identities we start with Pythagoras" Theorem:
Dividing through by c2 gives
This can be simplified to: ( Now, a/c is And So (a/c)2 + (b/c)2 = 1 can also be written: Note: sin2 θ means to find the sine of θ, then square the result, andsin 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 0.52992 + 0.84802 We get very close to 1 using only 4 decimal places. Try it on your calculator, you might get better results! sin2 θ = 1 − cos2 θ
## But Wait ... There is More!There are many more identities ... here are some of the more useful ones: ## Opposite Angle Identitiessin(−θ) = −sin(θ) cos(−θ) = cos(θ) tan(−θ) = −tan(θ) ## Double Angle Identities## Half Angle Identities
## Angle Sum and Difference IdentitiesNote 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) = cot(A B) = ## Triangle IdentitiesThere are also Triangle Identities which apply to all triangles (not just Right Angled Triangles) |