Trigonometry (Pure 2)
Reciprocal functions & identities
- Three reciprocal functions: $\sec\theta = \dfrac{1}{\cos\theta}$, $\csc\theta = \dfrac{1}{\sin\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$.
- Identities to know:
- $\sec^2\theta \equiv 1 + \tan^2\theta$,
- $\csc^2\theta \equiv 1 + \cot^2\theta$.
Practice
The secant function sec θ is equal to:
sec θ = 1/cos θ; csc θ = 1/sin θ; cot θ = 1/tan θ.
Practice
The identity sec²θ ≡ 1 + tan²θ. If tan θ = 2, what is sec²θ?
sec²θ = 1 + tan²θ = 1 + 2² = 1 + 4 = 5.
Angle formulae
- double angle: $\sin 2A = 2\sin A\cos A$, $\cos 2A = 2\cos^2 A - 1$.
- the R-formula: $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$, with $R = \sqrt{a^2 + b^2}$.
- Also the compound angle formulae for $\sin(A\pm B)$, $\cos(A\pm B)$.
Practice
For 3 sin θ + 4 cos θ = R sin(θ + α), what is R?
R = √(a² + b²) = √(3² + 4²) = √25 = 5.
You've got it
Key idea
- reciprocals: sec $=1/\cos$, csc $=1/\sin$, cot $=1/\tan$
- identities: $\sec^2\theta \equiv 1 + \tan^2\theta$, $\csc^2\theta \equiv 1 + \cot^2\theta$
- R-formula: $R = \sqrt{a^2 + b^2}$; double angle: $\sin 2A = 2\sin A\cos A$