Use compound angles and R-formulae

Retain the addition and double-angle formulae

Choose the formula that reduces the number of different trigonometric functions. To reduce an even power, use a cosine double-angle form and repeat if needed.

The source labels both formula groups “in MF27”. The paired signs in cosine and tangent switch as indicated by ∓:

$$\begin{aligned} \sin(A\pm B)&=\sin A\cos B\pm\cos A\sin B,\\ \cos(A\pm B)&=\cos A\cos B\mp\sin A\sin B,\\ \tan(A\pm B)&=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}. \end{aligned}$$

The double-angle formulae follow by taking equal angles:

$$\begin{aligned} \sin2A&=2\sin A\cos A,\\ \cos2A&=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1,\\ \tan2A&=\frac{2\tan A}{1-\tan^2A}. \end{aligned}$$

Tangent expressions require all terms and denominators involved to be defined. Example 20 retains the full triple-angle proof, correcting the source’s “sine terms only” callout to cosine terms. Example 21 retains both double-angle rearrangements that reduce cos⁴θ. Continue with the distinct Tutorial Question 14, which instead reduces sin⁴(θ/2).

Derive all four R-formulae

Expand the proposed shifted function and match its sine and cosine coefficients. Find the positive amplitude first, then the phase in the correct angle unit.

For positive a,b, write $a\cos x+b\sin x=R\cos(x-\alpha)$ with $R>0$ and acute α. Expanding gives

$$\begin{aligned} a\cos x+b\sin x&=R(\cos x\cos\alpha+\sin x\sin\alpha)\\&=R\cos\alpha\cos x+R\sin\alpha\sin x. \end{aligned}$$

Comparing coefficients gives (1) $R\cos\alpha=a$ and (2) $R\sin\alpha=b$. Divide (2) by (1) to obtain $\tan\alpha=b/a$, hence $\alpha=\tan^{-1}(b/a)$. Adding their squares gives

$$R^2\cos^2\alpha+R^2\sin^2\alpha=a^2+b^2 \ \Longrightarrow\ R^2=a^2+b^2 \ \Longrightarrow\ R=\sqrt{a^2+b^2}.$$

All four printed variants use this same R and α:

$$\begin{aligned} a\cos x+b\sin x&=R\cos(x-\alpha),\\ a\sin x+b\cos x&=R\sin(x+\alpha),\\ a\sin x-b\cos x&=R\sin(x-\alpha),\\ a\cos x-b\sin x&=R\cos(x+\alpha). \end{aligned}$$

Examples 22–23 preserve both extrema and the full shifted-bound chain. Their printed 56.3° and 22.6° phases are approximations; retain exact arctangents for identities, and use the same unit for the input angle and its phase. Continue with the distinct Tutorial Question 15.