AS/A-Level Mathematics - Trigonometric identities

Let's review the trigonometric identities in A-Level Maths!

Trigonometric Identities 

There are three reciprocal trigonometric functions, making a total of six including cosine, sine, and tangent.

The reciprocal cosine function is secant: secθ = 1/cosθ. 

The reciprocal sine function is cosecant, cscθ = 1/sinθ.

The reciprocal tangent function is cotangent, expressed two ways: cotθ = 1/tanθ or cotθ = cosθ/sinθ.

cos²θ + sin²θ = 1 

1 + tan²θ = sec²θ    

cot²θ + 1 = cosec²θ 

When asked to show by counter-example that a statement is false, sub a value in and show that LHS is not equal to RHS. 

Formulae for sin (A + B), cos (A + B), tan (A + B) 

Sin (A + B) = sin A cos B + cos A sin B 

Sin (A - B) = sin A cos B - cos A sin B 

Cos (A + B) = cos A cos B - sin A sin B 

Cos (A - B) = cos A cos B + sin A sin B 

Also, we see that sin(π/2-x) = cos(x), cos(π/2-x) = sin(x); 

that sin(x + π) = −sin(x), cos(x + π) = −cos(x); 

and that sin(π − x) = sin(x), cos(π − x) = −cos(x).

The formulas also give the tangent of a difference formula, for tan(alpha − beta).

Half-angle, double-angle

Double-angle identities derivation

Half-angle identities derivation

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Drafted by Eunice (Maths)

References:

https://www.onlinemathlearning.com/half-angle-examples.html

https://www.onlinemathlearning.com/double-angle-formula.html