CAIE A-Level Mathematics A2 3.3 Trigonometry Questions

Practise selecting compound-, double-angle, reciprocal and R-form identities to prove results, simplify expressions and solve every valid angle in a stated interval.

Exam points

select compound- or double-angle identities that reduce the expression to one useful form
write a sin x + b cos x as R sin(x ± α) or R cos(x ± α), finding exact R and α
solve the transformed equation and generate all interval solutions without introducing extra roots

Covered syllabus points

3.3.1—Reciprocal trig functions 3.3.2—Trig identities

[Maximum number: 2]

By sketching a suitable pair of graphs, show that the equation $\operatorname{cosec} x=1+\mathrm{e}^{-\frac{1}{2} x}$ has exactly two roots in the interval $0<x<\pi$ .

Sketch a relevant graph, e.g. $y=\operatorname{cosec} x$ $\operatorname{cosec} x$ , U shaped, roughly symmetrical about $x=\frac{\pi}{2}, y\left(\frac{\pi}{2}\right)=1$
and domain at least $\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$ .
Sketch a second relevant graph, e.g. $y=1+\mathrm{e}^{-\frac{1}{2} x}$ , and justify the
given statement
Exponential graph needs y(0)=2, negative gradient, always increasing, and $y(\pi)>1$
Needs to mark intersections with dots, crosses, or say roots at points of intersection, or equivalent

[Maximum number: 6]

(a)

Show that the equation $\tan^3x+2\tan2x-\tan x=0$ may be expressed as
$\tan^4x-2\tan^2x-3=0$
for $\tan x\ne0$ .

[ 3 ]

Use the double-angle formula:
$\tan2x=\frac{2\tan x}{1-\tan^2x}.$
For example,
$\tan^3x+\frac{4\tan x}{1-\tan^2x}-\tan x=0.$
Multiplying by $1-\tan^2x$ gives a correct equation in $\tan x$ , such as
$\tan^3x-\tan^5x+4\tan x-\tan x+\tan^3x=0.$
Since $\tan x\ne0$ , reduce correctly to
$\tan^4x-2\tan^2x-3=0.$

(b)

Hence solve the equation $\tan^3 2\theta+2\tan4\theta-\tan2\theta=0$ for $0<\theta<\pi$ . Give your answers in exact form.

[ 3 ]

Using part (a) with $x=2\theta$ :
$\tan^4 2\theta-2\tan^2 2\theta-3=0.$
Thus
$\tan^2 2\theta=3,\qquad \tan2\theta=\pm\sqrt3.$
For $0<\theta<\pi$ ,
$\theta=\frac{\pi}{6},\quad \frac{\pi}{3},\quad \frac{2\pi}{3},\quad \frac{5\pi}{6}.$
Exact answers only;
no others in the interval.

[Maximum number: 9]

(a)

Express $3 \sin x+2 \sqrt{2} \cos \left(x+\frac{1}{4} \pi\right)$ in the form $R \sin (x+\alpha)$ , where R>0 and $0<\alpha<\frac{1}{2} \pi$ . State the exact value of R and give $\alpha$ correct to 3 decimal places.

[ 4 ]

Use the correct expansion of $\cos \left(x+\frac{1}{4} \pi\right)$ to obtain $\sin x+2 \cos x$ $3 \sin x+2 \sqrt{2}\left(\frac{1}{\sqrt{2}} \cos x-\frac{1}{\sqrt{2}} \sin x\right)$ .

State $R=\sqrt{5}$

B1 FT

ISW FT their $a \sin x+b \cos x$ provided this
expression obtained by correct method.

Use correct trig formulae to find $\alpha$ $\alpha=\tan ^{-1}(b / a)$ from their $a \sin x+b \cos x$
or $\sin ^{-1}$ or $\cos ^{-1}$ provided this expression obtained
by correct method.
NB If $\cos \alpha=1$ and $\sin \alpha=2$ then M0 A0.

Obtain $\alpha=1.107$
3 d.p. CAO
Treat answer in degrees as a misread ( $63.435^{\circ}$ ) .

(b)

Hence solve the equation

6 \sin \frac{1}{2} \theta+4 \sqrt{2} \cos \left(\frac{1}{2} \theta+\frac{1}{4} \pi\right)=3

for $-4 \pi<\theta<4 \pi$ .

[ 5 ]

$\sin ^{-1}\left(\frac{1.5}{R}\right)$

B1 FT

Follow their R.

Use a correct method to obtain an un-simplified value of $\theta$ with their $\alpha$ $2\left(\sin ^{-1}\left(\frac{1.5}{R}\right)-\alpha\right)$ or $2\left(\pi-\sin ^{-1}\left(\frac{1.5}{R}\right)-\alpha\right)$ .

Obtain one correct answer e.g. -0.74 in the interval
Obtain second correct answer e.g. 2.60 (2.5986) or $4 \pi-0.74=11.8$
or $2.60-4 \pi=-9.97$ in the interval
If uses $1.11^{\circ}$ withhold first accuracy mark gained,
but allow rest of accuracy marks. Allow 2.6(0).

Obtain two more correct answers e.g. -9.97 and 11.8 and no others in the interval
Marking guidance:

Ignore answers outside the interval. Treat answers
in degrees as a misread.
$\left(-571.1^{\circ},-42.6^{\circ}, 148.9^{\circ}, 677.2^{\circ}\right)$ .

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)