CAIE A-Level Mathematics 1.5 Trigonometry Question Bank

Practise sketching trigonometric graphs, proving identities and solving equations in stated intervals using exact values, inverse functions, periodicity and all valid branches.

Exam points

sketch sine, cosine or tangent transformations with the correct period, range and key points
prove identities by transforming one side with reciprocal, quotient and Pythagorean identities
solve the reduced trigonometric equation and generate every solution in the stated interval

Covered syllabus points

1.5.1—Trig graphs 1.5.2—Exact trig values 1.5.3—Inverse trig notation 1.5.4—Trig identities 1.5.5—Trig equations

[Maximum number: 5]

(a)

Figure for Question (a) — CAIE A-Level Mathematics AS

The diagram shows the curve $y=k \cos \left(x-\frac{1}{6} \pi\right)$ where k is a positive constant and x is measured in radians. The curve crosses the x-axis at point A and B is a minimum point.

Find the coordinates of A and B.

[ 3 ]

State $\left(\frac{5}{3} \pi, 0\right)$ for point A

B1

Or exact equivalent.

Marking guidance:

Allow $x=\frac{5}{3} \pi$ or exact equivalent.

$x=\frac{19}{6} \pi$ for point B

B1

Or exact equivalent.
May be implied in coordinate or vector form.

y=-k for point B

B1

May be implied in coordinate or vector form.

(b)

Find the exact value of t that satisfies the equation

3 \sin ^{-1}(3 t)+2 \cos ^{-1}\left(\frac{1}{2} \sqrt{2}\right)=\pi .

Figure for Question (b) — CAIE A-Level Mathematics AS

[ 2 ]

Solve at least as far as $\sin ^{-1} 3 t=k \pi$ with correct value for $\cos ^{-1}\left(\frac{1}{2} \sqrt{2}\right)$

M1

Marking guidance:

Allow use of $\pi=3.14 \ldots$ .
Allow $\sin ^{-1} 3 t=30$ .

$\sin ^{-1} 3 t=\frac{1}{6} \pi$ and hence $t=\frac{1}{6}$

A1

Or exact equivalent.
Can use degrees if consistent.

[Maximum number: 8]

(a)

For the case where angle $B A C=\frac{1}{6} \pi$ radians, find k correct to 4 significant figures.

[ 5 ]

$\triangle A D E=\frac{1}{2}(k a)^{2} \sin \frac{\pi}{6}$
M1
Attempt to find the area of $\triangle A D E$ .
$\frac{1}{4} k^{2} a^{2}$
A1
OE.
Sector $A B C=\frac{1}{2} a^{2} \frac{\pi}{6}$
B1
$2 \times \frac{1}{4} k^{2} a^{2}=\frac{1}{2} a^{2} \frac{\pi}{6}$
M1
OE. For $2 \times \triangle A D E=$ sector A B C with at least one
correct area.
$k=\left(\sqrt{\frac{\pi}{6}}\right)=0.7236$
A1

(b)

For the general case in which angle $B A C=\theta$ radians, where $0<\theta<\frac{1}{2} \pi$ , it is given that $\frac{\theta}{\sin \theta}>1$ . Find the set of possible values of k.

[ 3 ]

$2 \times \frac{1}{2}(k a)^{2} \sin \theta=\frac{1}{2} a^{2} \theta$
M1
Condone omission of '2' or '1/2' on LHS for M1
only.
$k^{2}=\frac{\theta}{2 \sin \theta}$
A1
$k^{2}>\frac{1}{2}$ leading to $\frac{1}{\sqrt{2}}<k<1$
A1
OE. Accept $k>\frac{1}{\sqrt{2}}$ or k>0.707 (AWRT) or
0.707 (AWRT) <k<1 or $k>\sqrt{\frac{1}{2}}$ OE

All question bank results loaded

Covered syllabus points

1.5 Trigonometry question 1

Question (a)

Question (b)

1.5 Trigonometry question 2

Question (a)

Question (b)