Trigonometry: Solving Trigonometric Equations 1a

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Ben McGovern 4 years, 7 months ago

Created this as a copy of Trigonometry: Solving Trigonometric Equations 2.

Name	Status	Author	Last Modified
Trigonometry: Solving Trigonometric Equations 3	Ready to use	Ben McGovern	10/10/2022 13:12
Trigonometry: Solving Trigonometric Equations 2	Ready to use	Ben McGovern	10/10/2022 13:12
Trigonometry: Solving Trigonometric Equations 1a	Ready to use	Ben McGovern	10/10/2022 13:12
Trigonometry: Solving Trigonometric Equations 1b	Ready to use	Ben McGovern	07/09/2023 08:56

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Question statement

Solve $\sin(\var{n}x)=\var{a}$, for $x \in (0,\pi)$.

Give any introductory information the student needs.

			Name	Type	Generated Value

sol1a	number	0.103
sol2a	number	0.683
sol3a	number	1.674
sol4a	number	2.253

			Name	Type	Generated Value

n	integer	4
a	number	0.4
defs	list	[ [ "A", 4 ], [ "N", 0.4 ] ]
solnum	string	four
advice	string	<p>To find the third and forth
sol34	string	<p>To find the third and forth
solution	string	[ 0.103, 0.683, 1.674, 2.253 ]
sol1	number	0.1028792115
sol2	number	0.6825189519
sol3	number	1.6736755383
sol4	number	2.2533152787
solA	list	[ 0.103, 0.683 ]
solB	list	[ 0.103, 0.683, 1.674, 2.253 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
precround(sol1,3)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `number`

0.103

← Depends on:

sol1

→ Used by:

sol34
solA
solB

Advice

Parts

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1. Gap List of numbers
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(Give all possible solutions with your answers to 3 decimal places where necessary.)

Advice

To solve $\sin(\var{n}x)=\var{a}$, we need to rearrange the equation so that it is in terms of $x$:

\[ \begin{split} \sin(\var{n}x) &\, =\var{a} \\ \var{n}x &\,= \sin^{-1}(\var{a}) \\ x &\,= \frac{1}{\var{n}}\sin^{-1}(\var{a}). \end{split} \]

To check how many solutions we expect to have it can be worth sketching the function and marking when it is equal to $\var{a}$:

{geogebra_applet('https://www.geogebra.org/m/pqjfb3nf',defs)}

From the graph we can see that there are {solnum} solutions for $\sin(\var{n}x)=\var{a}$ {solnum} between $0$ and $\pi$.

The first solution is

\[ \begin{split} x &\,= \frac{1}{\var{n}}\sin^{-1}(\var{a}) \\ &\,=\var{sol1a} \text{ (3 d.p.)} \end{split} \]

Since $\sin(\var{n}x)$ is symmetrical between between $0$ and $\frac{\pi}{\var{n}}$, the second solution can be found by subtracting the first solution from $\frac{\pi}{\var{n}}$:

\[ \begin{split} x &\,= \frac{\pi}{\var{n}}-\var{sol1a} \\ &\,=\var{sol2a} \text{ (3 d.p.)}. \end{split} \]

{advice}

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Trigonometry: Solving Trigonometric Equations 1a

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Taxonomy: mathcentre

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Christian Lawson-Perfect 2 years, 7 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 4 years, 7 months ago

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Trigonometry: Solving Trigonometric Equations 1a

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 2 years, 7 months ago

Ben McGovern 3 years, 8 months ago

Ben McGovern 4 years, 7 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`