Trigonometry – Further Identities, Equations & Solutions (Cambridge 9709 P3)
1. Exact Values (required for the syllabus)
| Angle | sin θ | cos θ | tan θ |
|---|
| 0° (0) | 0 | 1 | 0 |
| 30° (π/6) | ½ | \(\sqrt3/2\) | 1/√3 |
| 45° (π/4) | \(\sqrt2/2\) | \(\sqrt2/2\) | 1 |
| 60° (π/3) | \(\sqrt3/2\) | ½ | √3 |
| 90° (π/2) | 1 | 0 | – |
2. Key Trigonometric Identities
All identities hold for any angle θ (radians or degrees). They are the tools you will use to transform and solve equations.
| Category | Identity |
|---|
| Reciprocal (six functions) |
$$\sin\theta=\frac1{\csc\theta},\quad\cos\theta=\frac1{\sec\theta},\quad\tan\theta=\frac1{\cot\theta}$$
$$\csc\theta=\frac1{\sin\theta},\quad\sec\theta=\frac1{\cos\theta},\quad\cot\theta=\frac1{\tan\theta}$$
|
| Pythagorean |
$$\sin^{2}\theta+\cos^{2}\theta=1$$
$$1+\tan^{2}\theta=\sec^{2}\theta$$
$$1+\cot^{2}\theta=\csc^{2}\theta$$
|
| Co‑function (π/2‑shift) |
$$\sin\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)=\cos\theta,\qquad
\cos\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)=\sin\theta$$
$$\tan\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)=\cot\theta,\qquad
\cot\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)=\tan\theta$$
$$\sec\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)=\csc\theta,\qquad
\csc\!\Bigl(\tfrac{\pi}{2}-\theta\Bigr)=\sec\theta$$
|
| Sum‑to‑Product |
$$\sin A\pm\sin B=2\sin\frac{A\pm B}{2}\cos\frac{A\mp B}{2}$$
$$\cos A+\cos B=2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$$
$$\cos A-\cos B=-2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$$
|
| Product‑to‑Sum |
$$\sin A\sin B=\tfrac12\bigl[\cos(A-B)-\cos(A+B)\bigr]$$
$$\cos A\cos B=\tfrac12\bigl[\cos(A-B)+\cos(A+B)\bigr]$$
$$\sin A\cos B=\tfrac12\bigl[\sin(A+B)+\sin(A-B)\bigr]$$
|
| Double‑angle |
$$\sin2\theta=2\sin\theta\cos\theta$$
$$\cos2\theta=\cos^{2}\theta-\sin^{2}\theta
=2\cos^{2}\theta-1
=1-2\sin^{2}\theta$$
$$\tan2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}$$
|
| Triple‑angle |
$$\sin3\theta=3\sin\theta-4\sin^{3}\theta$$
$$\cos3\theta=4\cos^{3}\theta-3\cos\theta$$
$$\tan3\theta=\frac{3\tan\theta-\tan^{3}\theta}{1-3\tan^{2}\theta}$$
|
| R sin θ + b cos θ (or R cos θ + a sin θ) form |
$$R\sin\theta+b\cos\theta
=\sqrt{R^{2}+b^{2}}\;\sin\!\bigl(\theta+\alpha\bigr),$$
where $$\alpha=\arctan\!\Bigl(\frac{b}{R}\Bigr)$$
(choose the quadrant of α so that
$\cos\alpha=R/\sqrt{R^{2}+b^{2}}$ and $\sin\alpha=b/\sqrt{R^{2}+b^{2}}$).
The analogous result for $R\cos\theta+a\sin\theta$ is obtained by shifting the phase by $-\alpha$.
|
3. General Solutions of the Basic Trigonometric Equations
When the question asks for “all solutions”, give the principal solution(s) and then add the appropriate multiple of the function’s period. Remember:
- Period of $\sin$, $\cos$, $\sec$, $\csc$ → $2\pi$ (or $360^\circ$)
- Period of $\tan$, $\cot$ → $\pi$ (or $180^\circ$)
| Equation | General solution |
|---|
| $\sin\theta = k$ (|k|≤1) |
$$\theta = \arcsin k + 2n\pi\quad\text{or}\quad
\theta = \pi - \arcsin k + 2n\pi,\; n\in\mathbb Z$$ |
| $\cos\theta = k$ (|k|≤1) |
$$\theta = \arccos k + 2n\pi\quad\text{or}\quad
\theta = -\arccos k + 2n\pi,\; n\in\mathbb Z$$ |
| $\tan\theta = k$ |
$$\theta = \arctan k + n\pi,\; n\in\mathbb Z$$ |
| $\cot\theta = k$ |
$$\theta = \arccot k + n\pi,\; n\in\mathbb Z$$ |
| $\sec\theta = k$ (|k|≥1) |
Reduce to $\cos\theta = 1/k$ and use the cosine formula above. |
| $\csc\theta = k$ (|k|≥1) |
Reduce to $\sin\theta = 1/k$ and use the sine formula above. |
4. Systematic Procedure for Solving Trigonometric Equations
- Simplify. Use identities to rewrite the equation so that at most one trig function (or a simple product) appears.
- Factor / Apply sum‑to‑product. Convert sums to products when possible; factor common terms.
- Isolate the function. Set each factor equal to zero (or solve a simple equation such as $\sin\theta = k$).
- Find principal solutions. Work in the interval required by the question (commonly $0\le\theta<2\pi$ or $0^\circ\le\theta<360^\circ$).
- Write the general solution. Add $2\pi$ (or $360^\circ$) for $\sin,\cos,\sec,\csc$ and $\pi$ (or $180^\circ$) for $\tan,\cot$.
- Check for extraneous roots. Substitute each answer back into the original equation, especially after squaring, multiplying by a trig function, or using reciprocal identities.
5. Representative Worked Examples (full syllabus coverage)
Example 1 – Quadratic in $\cos\theta$ (uses double‑angle identity)
Solve $$2\cos^{2}\theta-3\cos\theta+1=0,\qquad 0\le\theta<2\pi.$$
- Let $x=\cos\theta$. Equation becomes $2x^{2}-3x+1=0$.
- Factor: $(2x-1)(x-1)=0\;\Longrightarrow\;x=\frac12\text{ or }x=1$.
- Recover $\theta$ using the cosine table:
- $\cos\theta=\frac12\;\Rightarrow\;\theta=\frac{\pi}{3},\;\frac{5\pi}{3}$.
- $\cos\theta=1\;\Rightarrow\;\theta=0$.
- Answer: $\displaystyle\theta=0,\;\frac{\pi}{3},\;\frac{5\pi}{3}$.
Example 2 – Sum‑to‑Product (sin θ + sin 3θ)
Solve $$\sin\theta+\sin3\theta=0,\qquad 0\le\theta<2\pi.$$
- Apply sum‑to‑product: $\sin\theta+\sin3\theta=2\sin2\theta\cos\theta$.
- Product zero ⇒ $\sin2\theta=0$ or $\cos\theta=0$.
- Solutions:
- $\sin2\theta=0\;\Rightarrow\;2\theta=n\pi\;\Rightarrow\;\theta=\frac{n\pi}{2}$.
- $\cos\theta=0\;\Rightarrow\;\theta=\frac{\pi}{2}+n\pi$.
Within $0\le\theta<2\pi$: $\theta=0,\;\frac{\pi}{2},\;\pi,\;\frac{3\pi}{2}$.
Example 3 – $R\sin\theta+b\cos\theta$ form
Solve $$3\sin\theta+4\cos\theta=2,\qquad 0\le\theta<2\pi.$$
- Compute $R=\sqrt{3^{2}+4^{2}}=5$, $\displaystyle\alpha=\arctan\!\Bigl(\frac{4}{3}\Bigr)\approx0.9273\text{ rad}$ (53.13°).
$$3\sin\theta+4\cos\theta=5\sin(\theta+\alpha).$$
- Equation becomes $5\sin(\theta+\alpha)=2\;\Longrightarrow\;\sin(\theta+\alpha)=\frac25.$
- Principal solutions for $\phi=\theta+\alpha$:
$$\phi_{1}= \arcsin\frac25\approx0.4115,\qquad
\phi_{2}= \pi-\arcsin\frac25\approx2.7301.$$
- Subtract $\alpha$:
- $\theta_{1}=0.4115-0.9273\approx-0.5158\;\Rightarrow\;\theta_{1}=2\pi-0.5158\approx5.7674\text{ rad}.$
- $\theta_{2}=2.7301-0.9273\approx1.8028\text{ rad}.$
- Answer (rad): $\theta\approx1.803,\;5.767$
(≈ 103.3° and 330.3°).
Example 4 – Equation involving $\sec\theta$ (reciprocal identity)
Solve $$\sec\theta-2\cos\theta=0,\qquad 0\le\theta<2\pi.$$
- Replace $\sec\theta$ by $1/\cos\theta$:
$$\frac1{\cos\theta}-2\cos\theta=0\;\Longrightarrow\;1-2\cos^{2}\theta=0.$$
- Thus $\cos^{2}\theta=\tfrac12\;\Rightarrow\;\cos\theta=\pm\frac1{\sqrt2}.$
- Using the cosine table:
- $\cos\theta=+\frac1{\sqrt2}\;\Rightarrow\;\theta=\frac{\pi}{4},\;\frac{7\pi}{4}$.
- $\cos\theta=-\frac1{\sqrt2}\;\Rightarrow\;\theta=\frac{3\pi}{4},\;\frac{5\pi}{4}$.
- Answer: $\displaystyle\theta=\frac{\pi}{4},\;\frac{3\pi}{4},\;\frac{5\pi}{4},\;\frac{7\pi}{4}$.
Example 5 – Quadratic in $\cot\theta$ (using reciprocal of $\tan$)
Solve $$2\cot^{2}\theta-3\cot\theta-2=0,\qquad 0\le\theta<2\pi.$$
- Let $x=\cot\theta$. $2x^{2}-3x-2=0$.
- Factor: $(2x+1)(x-2)=0\;\Rightarrow\;x=-\tfrac12\text{ or }x=2$.
- Return to $\theta$:
- $\cot\theta=2\;\Rightarrow\;\theta=\arccot2\approx0.4636\text{ rad}$ or $\theta=\pi+0.4636\approx3.6052\text{ rad}$.
- $\cot\theta=-\tfrac12\;\Rightarrow\;\theta=\pi-\arccot\tfrac12\approx2.6780\text{ rad}$ or $\theta=2\pi-\arccot\tfrac12\approx5.8195\text{ rad}$.
- Answer (rad): $\theta\approx0.464,\;2.678,\;3.605,\;5.820$
(≈ 26.6°, 153.4°, 206.6°, 333.4°).
Example 6 – Solving a mixed‑function equation (product‑to‑sum)
Solve $$\sin2\theta\cos3\theta=0,\qquad 0\le\theta<2\pi.$$
- Product zero ⇒ $\sin2\theta=0$ or $\cos3\theta=0$.
- $\sin2\theta=0\;\Rightarrow\;2\theta=n\pi\;\Rightarrow\;\theta=\frac{n\pi}{2}$.
- $\cos3\theta=0\;\Rightarrow\;3\theta=\frac{\pi}{2}+n\pi\;\Rightarrow\;\theta=\frac{\pi}{6}+\frac{n\pi}{3}$.
- List values in $[0,2\pi)$:
- From $\theta=\frac{n\pi}{2}$: $0,\;\frac{\pi}{2},\;\pi,\;\frac{3\pi}{2}$.
- From $\theta=\frac{\pi}{6}+\frac{n\pi}{3}$: $\frac{\pi}{6},\;\frac{\pi}{2},\;\frac{5\pi}{6},\;\frac{7\pi}{6},\;\frac{3\pi}{2},\;\frac{11\pi}{6}$.
- Combine and remove duplicates →
$\displaystyle\theta=0,\;\frac{\pi}{6},\;\frac{\pi}{2},\;\frac{5\pi}{6},\;\pi,\;\frac{7\pi}{6},\;\frac{3\pi}{2},\;\frac{11\pi}{6}$.
6. Quick Reference – Periods & Principal Ranges
- sin θ, cos θ, sec θ, csc θ: period $2\pi$; principal range $[0,2\pi)$.
- tan θ, cot θ: period $\pi$; principal range $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ for $\tan$ and $(0,\pi)$ for $\cot$.
- When using inverse functions, always note the principal value and then add the appropriate multiple of the period.
7. Checklist Before Finalising Answers
- Have you reduced the equation to the simplest possible trig form?
- Did you consider the domain of the original equation (e.g., $\cos\thetaeq0$ when a denominator is present)?
- Are all solutions within the requested interval?
- Have you added the correct period for the general solution?
- Did you substitute each solution back into the original equation to confirm it is not extraneous?