CAIE A-Level Mathematics AS 2.3 Trigonometry Questions

Practise reciprocal trigonometric functions, identities and R-formula transformations to solve equations and retain every valid angle inside specified degree or radian intervals.

Exam points

replace sec, cosec and cot with reciprocal or quotient forms and apply their identities
express a sin x + b cos x as R sin(x ± α) or R cos(x ± α) with correct quadrant
solve the reduced equation and keep all solutions lying in the specified interval

Covered syllabus points

2.3.1—Reciprocal trig functions 2.3.2—Trig identities

[Maximum number: 6]

It is given that $3 \sin 2 \theta=\cos \theta$ where $\theta$ is an angle such that $0^{\circ}<\theta<90^{\circ}$ .

(a)

Find the exact value of $\sin \theta$ .

[ 2 ]

Use $\sin 2 \theta=2 \sin \theta \cos \theta$
Obtain $\sin \theta=\frac{1}{6}$

(b)

Find the exact value of $\sec \theta$ .

[ 2 ]

Use correct identity or identities to find value of $\sec \theta$
Obtain $\frac{6}{\sqrt{35}}$ or exact equivalent

(c)

Find the exact value of $\cos 2 \theta$ .

[ 2 ]

Use correct identity or identities to find value of $\cos 2 \theta$
Obtain $\frac{17}{18}$ or exact equivalent

[Maximum number: 8]

The expression $\mathrm{f}(\theta)$ is defined by $\mathrm{f}(\theta)=12 \sin \theta \cos \theta+16 \cos ^{2} \theta$ .

(a)

Express $\mathrm{f}(\theta)$ in the form $R \cos (2 \theta-\alpha)+k$ , where $R>0,0<\alpha<\frac{1}{2} \pi$ and k is a constant. State the values of R and k, and give the value of $\alpha$ correct to 4 significant figures.

[ 5 ]

Use correct identity for $\sin 2 \theta$ or $\cos 2 \theta$ (or both)
Obtain $6 \sin 2 \theta+8 \cos 2 \theta(+8)$
State R=10

B1 FT

FT their $a \sin 2 \theta+b \cos 2 \theta$ form.

Use appropriate trigonometry to find $\alpha$ using their $a \sin 2 \theta+b \cos 2 \theta$ form
Marking guidance:

Allow 0.927.

Obtain $10 \cos (2 \theta-0.6435)+8$

(b)

Find the smallest positive value of $\theta$ satisfying the equation $\mathrm{f}(\theta)=17$ .

[ 3 ]

State or imply $\cos (2 \theta-0.6435)=0.9$

B1 FT

FT their $R, \alpha$ and k provided RHS less than 1 .

Carry out correct process to find any positive value of $\theta$
0.547

Obtain 0.0962
AWRT

[Maximum number: 3]

The polynomial p(x) is defined by

\mathrm{p}(x)=6 x^{3}+a x^{2}-4 x-3,

where a is a constant. It is given that ( x+3 ) is a factor of p(x).

Hence solve the equation $\mathrm{p}(\operatorname{cosec} \theta)=0$ for $0^{\circ}<\theta<360^{\circ}$ .

Use the factorisation from part (b) with $x=\cosec\theta$ :
$(\cosec\theta+3)(3\cosec\theta+1)(2\cosec\theta-1)=0.$

Attempt a solution of
$\sin\theta=k,\qquad -1\le k\le1.$
The valid equation is
$\cosec\theta=-3 \quad\Rightarrow\quad \sin\theta=-\frac13.$ $\theta=199.5^\circ.$ $\theta=340.5^\circ.$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question 3