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Trigonometric Ratios and Identities · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SDefinitions of sine, cosine, and tangents7 marksPaper 1~11 min

A ladder of unknown length leans against a vertical wall. The base of the ladder is

2\,\text{m}

from the wall, and the ladder makes angle of

60°

with the horizontal ground.

(a)

Draw and label a right-angled triangle representing this situation, identifying the

2\,\text{m}

base, the angle of

60°

, and the length of the ladder

l

. [1 mark]

(b)

Calculate the length of the ladder

l

. [2 marks]

(c)

A second ladder of length

5\,\text{m}

leans against the same wall with its base also

2\,\text{m}

from the wall. Determine the height reached by the second ladder and evaluate whether it reaches more than

1\,\text{m}

higher up the wall than the first ladder. [4 marks]

Solutions

2FoundationSAQ-SDefinitions of sine, cosine, and tangents5 marksPaper 1~8 min

In a right-angled triangle

PQR

, angle

Q = 90°

PQ = 5\,\text{cm}

, and

PR = 13\,\text{cm}

(a)

State the value of

\sin \angle PRQ

. [1 mark]

(b)

Show that

QR = 12\,\text{cm}

. [2 marks]

(c)

Hence find the exact value of

\sin 2\angle RPQ

. [2 marks]

Solutions

3MasterySAQ-SDefinitions of sine, cosine, and tangents5 marksPaper 1~8 min

A vertical pole of height

12\,\text{m}

stands on horizontal ground. From a point

P

on the ground, the angle of elevation of the top of the pole is

\theta

, where

\tan\theta = \dfrac{3}{4}

(a)

Show that

\sin\theta = \dfrac{3}{5}

. [2 marks]

(b)

Hence find the straight-line distance from

P

to the top of the pole. [3 marks]

Solutions

4ChallengeSAQ-LDefinitions of sine, cosine, and tangents8 marksPaper 1~12 min

A water wheel of radius

5

metres rotates anticlockwise at a constant speed. The point

P

is at the lowest point of the wheel at time

t = 0

seconds. The height

h

metres of

P

above the riverbed is modelled by

h(t) = a + b\cos(ct)

where

a

b

, and

c

are positive constants,

t

is measured in seconds, and the centre of the wheel is

6

metres above the riverbed.

(a)

Write down the values of

a

and

b

. [2 marks]

(b)

Given that the wheel completes one full revolution every

12

seconds, find the value of

c

. [2 marks]

(c)

Find the exact values of

t

in the interval

0 \leq t \leq 24

for which

h(t) = 8.5

metres, and justify that no other solutions exist in this interval. The following identities may be useful:

\cos(\theta) = \cos(-\theta)

\quad \cos(\theta) = \cos(\theta + 2\pi k)

for

k \in \mathbb{Z}

[4 marks]

Solutions

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