Advanced identities and harmonic forms

What you’ll learn

• Use reciprocal, compound-angle and double-angle identities.
• Rewrite sine–cosine combinations in a single harmonic form.
• Use inverse trigonometric principal ranges correctly.

Understand the idea

Secant, cosecant and cotangent are the reciprocals of cosine, sine and tangent where defined. Their asymptotes occur where the relevant denominator is zero. The principal ranges of $\arcsin$, $\arccos$ and $\arctan$ select a single inverse value; an equation may still have additional solutions in its stated interval.

Compound-angle identities connect the trigonometric values of a sum or difference to those of its parts. Setting both angles equal gives $\sin2x=2\sin x\cos x$ and $\cos2x$ $=\cos^2x-\sin^2x$ $=1-2\sin^2x$ $=2\cos^2x-1$. For tangent, $\tan(A+B)=(\tan A+\tan B)/(1-\tan A\tan B)$ where defined. Choose the form that reduces the number of different functions in a calculation.

An expression $a\cos x+b\sin x$ can be written as $R\cos(x-\alpha)$ by matching coefficients. The amplitude $R=\sqrt{a^2+b^2}$ gives the maximum and minimum immediately. Determine the quadrant of $\alpha$ using the signs of both coefficients, not only a calculator's inverse tangent.

Key formulae

$$\sec^2x=1+\tan^2x,\qquad \csc^2x=1+\cot^2x$$

$$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$$

$$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$$

$$a\cos x+b\sin x=R\cos(x-\alpha),\quad R\cos\alpha=a,\ R\sin\alpha=b$$

Worked example

Write $3\cos x+4\sin x$ as $R\cos(x-\alpha)$, with $R>0$ and $0<\alpha<90^\circ$, and state its maximum.

1. Expand $R\cos(x-\alpha)=R\cos\alpha\cos x+R\sin\alpha\sin x$.
2. Match coefficients: $R\cos\alpha=3$ and $R\sin\alpha=4$, giving $R=5$.
3. Both coefficients are positive, so $\alpha$ is in the first quadrant and $\alpha=\tan^{-1}(4/3)$.

Answer: $5\cos(x-53.130\ldots^\circ)$; maximum $5$.