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

1FoundationSAQ-SUnit circle and angle measurements5 marksPaper 1~8 min

A point

P

lies on the unit circle such that the angle

\theta

between the positive

x

-axis and the line

OP

\dfrac{5\pi}{6}

radians, measured anticlockwise.

(a)

State the coordinates of

P

. [2 marks]

(b)

Using your answer to part (a) and an appropriate compound-angle identity, find the exact value of

\cos\left(\dfrac{5\pi}{3}\right)

. [3 marks]

Solutions

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

A particle moves such that its displacement,

s

metres, from a fixed point

O

at time

t

seconds is given by

s(t) = 5\cos\!\left(\frac{\pi t}{3}\right) + 12\sin\!\left(\frac{\pi t}{3}\right), \quad t \geq 0.

(a)

Show that

s(t)

can be written in the form

s(t) = R\cos\!\left(\dfrac{\pi t}{3} - \alpha\right)

, where

R > 0

and

0 < \alpha < \dfrac{\pi}{2}

. State the value of

R

and the exact value of

\alpha

in the form

\arctan(k)

. [3 marks]

(b)

Hence find the first time

t > 0

at which the particle is at

O

, giving your answer in the form

a + \dfrac{b}{\pi}\arctan(k)

, where

a, b, k

are constants to be determined. [2 marks]

Solutions

3ChallengeSAQ-LDefinitions of sine, cosine, and tangents8 marksPaper 1~12 min

A surveyor measures the angle of elevation to the top of a vertical cliff from two points on level ground. From point

A

, the angle of elevation is

\dfrac{\pi}{6}

radians. From point

B

, which is

120

metres closer to the base of the cliff, the angle of elevation is

\dfrac{\pi}{3}

radians. Let

d

be the horizontal distance from

B

to the base of the cliff, and let

h

be the height of the cliff.

(a)

Show that

h = d\sqrt{3}

and hence determine the exact value of

h

. [4 marks]

(b)

The surveyor now considers a general scenario in which the angle of elevation from

A

\alpha

radians, where

0 < \alpha < \dfrac{\pi}{3}

, the angle from

B

remains

\dfrac{\pi}{3}

, and the distance

AB

remains

120

m. Show that the height of the cliff is given by

h = \frac{120\sqrt{3}\tan\alpha}{\sqrt{3} - \tan\alpha}

and hence deduce, with justification, whether

h

is an increasing or decreasing function of

\alpha

on the given domain. [4 marks]

Solutions

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