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Trigonometry

Trigonometric Identities

If the hypotenuse of a right angled triangle is 1 unit, then by the definitions of the sine and the cosine the opposite side is sin θ and the adjacent side is cos θ.

By the Pythagoras Theorem,

This is usually written as 

Dividing the above identity by cos²θ, we get

Divide the identity by sin ²θ, we get

The three formulas given in the boxes above are known as trigonometric identities as they provide relationships between trig functions. These identities are useful in solving a variety of problems. There are more trig identities that are not as easy to establish.

For any angle θ:

cos²θ + sin²θ = 1,

1 + tan²θ = sec²θ,

cot²θ + 1 = cosec²θ,

sin (–θ) = –sin θ,

cos (–θ) = cos θ,

tan (–θ ) = –tan θ,

cos 2θ = cos²θ– sin²θ = 2 cos²θ– 1 = 1 – 2 sin²θ,

sin 2θ = 2 sin θ cos θ,

.

 

For any angles A and B:

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) = ,

sin A + sin B = 2 sin (A + B) cos (A – B),

sin A – sin B = 2 sin (A – B) cos (A + B),

cos A + cos B = 2 cos (A + B) cos (A – B),

cos A – cos B = 2 sin (A + B) sin (B – A).

These identities can be used to simplify algebraic expressions to solve them or differentiate or integrate them.

<< Finding the Length of a Side in Right Angled Triangles  | Trigonometry Index | Using the Value of One Trig Function to Find Another >>