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Trigonometric Functions · Paper 1 — Free Maths AA HL Practice Questions

1FoundationSAQ-SSine, cosine, and tangent functionss8 marksPaper 1~12 min

The height,

h

metres, of the water above a fixed point a harbour entrance is modelled by

h(t) = 3 + 4\sin\!\left(\frac{\pi t}{6}\right)

where

t

is the time in hours after midnight,

0 \leq t \leq 24

(a)

State the amplitude of

h(t)

. [1 mark]

(b)

Calculate the height of the water at 2:00 am. [2 marks]

(c)

A boat requires a minimum water height of

5\,\text{m}

to enter the harbour safely. Determine the values of

t

, in the interval

0 \leq t \leq 12

, for which the boat may enter the harbour. (Give your answers as exact values.) [5 marks]

Solutions

2MasterySAQ-STrigonometric identities and equations5 marksPaper 1~8 min

Consider the equation

2\cos^2 x + 3\sin x = 3

, for

0 \leq x \leq 2\pi

(a)

Show that this equation can be written as

2\sin^2 x - 3\sin x + 1 = 0

. [2 marks]

(b)

Hence solve

2\cos^2 x + 3\sin x = 3

for

0 \leq x \leq 2\pi

, giving your answers in exact form. [3 marks]

Solutions

3ChallengeSAQ-LSine, cosine, and tangent functionss8 marksPaper 1~12 min

Consider the function

f(x) = \sin x + \cos x

defined for

x \in \mathbb{R}

(a)

Show that

f(x)

can be written in the form

R\sin(x + \alpha)

, where

R > 0

and

0 \leq \alpha < 2\pi

, and determine the exact values of

R

and

\alpha

. [3 marks]

(b)

Hence, determine the exact solutions of the equation

f(x) = 1

for

0 \leq x \leq 2\pi

. [2 marks]

(c)

The function

g

is defined by

g(x) = \dfrac{1}{f(x)}

for

x \in [0, 2\pi]

, excluding values where

f(x) = 0

. State the

x

-values in

[0, 2\pi]

where

g(x)

is undefined, and hence determine the range of

g

on the largest open interval within

[0, 2\pi]

on which

g

is continuous. [3 marks]

Solutions

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