Trigonometry (Pure 2)
| English | Chinese | Pinyin |
|---|---|---|
| R-formula | 辅助角公式 | fǔ zhù jiǎo gōng shì |
| reciprocal functions | 倒数函数 | dào shǔ hán shù |
| double angle | 二倍角 | èr bèi jiǎo |
The hidden patterns in waves
- Radio signals, sound waves, and alternating current all combine sine and cosine terms.
- The R-formula 辅助角公式 lets you merge $a\sin\theta + b\cos\theta$ into a single sine wave — essential for finding maximum amplitude.
Reciprocal functions 倒数函数
- Three reciprocal functions: $\sec\theta = \dfrac{1}{\cos\theta}$, $\csc\theta = \dfrac{1}{\sin\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$.
- These appear in calculus and when solving trig equations.
The graph of sec θ = 1/cos θ has asymptotes wherever cos θ = 0
The reciprocal functions: $\sec$, $\csc$, and $\cot$ are simply $\dfrac{1}{\cos}$, $\dfrac{1}{\sin}$, and $\dfrac{1}{\tan}$.
Worked example. If $\cos\theta = \dfrac{3}{5}$, then $\sec\theta = \dfrac{5}{3}$.
The unit circle
(cos θ, sin θ)
Every trig identity comes from this circle — watch how sin θ and cos θ relate as θ turns.
The secant function sec θ is equal to:
sec θ = 1/cos θ; csc θ = 1/sin θ; cot θ = 1/tan θ.
Pythagorean identities for reciprocals
- Identities to know:
- $\sec^2\theta \equiv 1 + \tan^2\theta$,
- $\csc^2\theta \equiv 1 + \cot^2\theta$.
Don't confuse these. $\sec^2\theta \equiv 1 + \tan^2\theta$ (not $\sec^2\theta = 1 - \tan^2\theta$). The sign is plus, derived from dividing $\sin^2\theta + \cos^2\theta \equiv 1$ by $\cos^2\theta$.
The identity sec²θ ≡ 1 + tan²θ. If tan θ = 2, what is sec²θ?
sec²θ = 1 + tan²θ = 1 + 2² = 1 + 4 = 5.
sec²θ ≡ 1 − tan²θ.
The correct identity is sec²θ ≡ 1 + tan²θ (plus, not minus).
Double angle 二倍角 formulae
- Double angle: $\sin 2A = 2\sin A\cos A$, $\cos 2A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$.
- These are essential for integrating powers of sin and cos.
Using the double angle formula, sin 60° = 2 sin 30° cos 30°. Evaluate (2 dp).
2 × 0.5 × (√3/2) = √3/2 ≈ 0.87.
The R-formula
- The R-formula: $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$, with $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \dfrac{b}{a}$.
- This converts a sum of sin and cos into a single wave — making it easy to find the maximum value ($R$) and where it occurs.
- Use the reciprocal functions secant, cosecant and cotangent.
For 3 sin θ + 4 cos θ = R sin(θ + α), what is R?
R = √(a² + b²) = √(3² + 4²) = √25 = 5.
The maximum value of 5 sin θ + 12 cos θ is:
R = √(5² + 12²) = √(25 + 144) = √169 = 13. Maximum value = R.
You've got it
- 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: $\sin 2A = 2\sin A\cos A$