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VCE 12 Methods 2023

3.06 General solutions

Worksheet
General solutions
1

Consider the equation \sin 2 x = \sqrt{\dfrac{3}{2}} - \cos 2 x.

a

Find all of the solutions to the equation in the interval 0 \leq x < 2 \pi.

b

Given that n represents an integer, find an expression for the general solutions to the equation \sin 2 x = \sqrt{\dfrac{3}{2}} - \cos 2 x.

2

Find the general solution of the following equations, using n to represent an integer:

a

\sin x = \dfrac{1}{2}

b

\cos x = - \dfrac{\sqrt{3}}{2}

c

\sin x = \dfrac{\sqrt{3}}{2}

d

\tan x =-\sqrt{3}

3

Find the general solution of the following equations, using n to represent an integer:

a

2 \sin x = -\sqrt{3}

b

\sqrt{3} \tan \left(\dfrac{x}{2}\right) = - 3

c

\sin 2 x = \dfrac{1}{\sqrt{2}}

d

2 \sin 3 x - \sqrt{2} = 0

e

\sin x = - \cos x

f

\cos \left(\dfrac{x}{2}\right) = 1 - \cos \left(\dfrac{x}{2}\right)

4

Find the general solution of the following equations, using n to represent an integer:

a

2 \sin ^{2} x = 1

b

\tan ^{2}3x = \dfrac{1}{3}

c

\cos ^{2}\left(\dfrac{x}{2}\right) - 1 = 0

d

\sin ^{2}\left(\dfrac{x}{2}\right) - \dfrac{3}{4} = 0

5

Find the general solution of the following equations, using n to represent an integer:

a

\cos \left(x + \dfrac{\pi}{4}\right) = - \dfrac{1}{\sqrt{2}}

b
\sin \left(x - \dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{2}}
c
\cos \left(x + \dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{2}
d
\sqrt{2} \cos \left(x - \dfrac{\pi}{3}\right) = 1
e
\tan \left(x + \dfrac{\pi}{3}\right) = \sqrt{3}
f
\sin \left( 3 x - \dfrac{\pi}{2}\right) = - 1
6

Find the general solution of the following equations, using n to represent an integer:

a

\cos ^{2} x + \cos x = 0

b

\cos x \tan x = \cos x

c
\sin ^2 \left(x - \dfrac{\pi}{4} \right) - 1=0
d
12\cos ^2 \left(2x - \dfrac{\pi}{4} \right) - 3=0
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Outcomes

U34.AoS2.5

solution of literal equations and general solution of equations involving a single parameter

U34.AoS2.9

apply a range of analytical, graphical and numerical processes (including the algorithm for Newton’s method), as appropriate, to obtain general and specific solutions (exact or approximate) to equations (including literal equations) over a given domain and be able to verify solutions to a particular equation or equations over a given domain

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