Trigonometry (Pure 2)

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 $\dfrac{1}{\tan}$.

Pythagorean identities for reciprocals

• Identities to know:
  • $\sec^2\theta \equiv 1 + \tan^2\theta$,
  • $\csc^2\theta \equiv 1 + \cot^2\theta$.

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.

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.

You've got it