Apply the sine rule

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 4 months ago

Created this.

Name	Status	Author	Last Modified
Apply the sine rule	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Robert's copy of Apply the sine rule	draft	Robert Pullan	18/07/2016 23:56

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Question statement

Suppose that $\Delta ABC$ is a triangle with $C> \dfrac{\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\Delta ABC$ ).

Given the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.

Give any introductory information the student needs.

			Name	Type	Generated Value

bb11	number	0.7300926536
cc1	number	1.9230926536
cc0	number	1.9231
b0	integer	27
cc2	number	1.923
check	number	-0.0000073464
q0	number	0.7451523546
s2	number	0.4688602824
s0	number	0.4693018599
u1	number	0.9386037199
u0	number	0.9385802965
u2	number	0.9386147979
aa1	number	0.4884926536
aa0	number	0.4885
aa2	number	0.488
cc01	number	1.923
bb21	number	0.730
a0	integer	19
c0	integer	38
p0	number	0.8830409357
r0	number	-0.3450292398
t2	number	0.666869635
t0	number	0.666869635
t21	number	0.666869635
c01	integer	35
c02	integer	44
bb2	number	0.730
bb1	number	0.7299926536
bb0	number	0.7300

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Data type

Value

xxxxxxxxxx
 
pi-aa0-cc01

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Description

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

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Generated value: `number`

0.7300926536

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This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
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1. Step Information only

Gap-fill

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$A=\var{AA0}$ , $a=\var{a0}$ , $c=\var{c0}$

Angle $C=$ Gap 0.

Angle $B=$ Gap 1.

Side length $b=$ Gap 2.

Advice

We use the Sine Rule to find $C$ : $\dfrac{a}{\sin A}=\dfrac{c}{\sin C}$ . Thus $\sin C=\dfrac{c \sin A}{a}=\dfrac{\var{c0}* \var{s0}}{\var{a0}}=\var{c0*s0/a0}$ . To find $C$ we need to calculate $\sin^{-1} (\var{c0*s0/a0})$ , calculating the angle between $\dfrac{\pi}{2}$ and $\pi$ , so $C=\var{cc01}$ (to 3 decimal places). [Using a calculator, $\sin^{-1} (\var{c0*s0/a0})$ returns $\var{arcsin(u1)}$ , which lies between $0$ and $\dfrac{\pi}{2}$ , so the value we need is $\pi-\var{arcsin(u1)}=\var{pi-arcsin(u1)}$ .]

Since $A+B+C=\pi$ , we calculate $B=\pi-A-C=\var{bb11}$ . To 3dp, this gives $\var{bb21}$ .

We use the Sine Rule to find $b$ : $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$ . Thus $b=\dfrac{a \sin B}{\sin A}=\dfrac{\var{a0}* \var{t21}}{\var{s0}}=\var{a0*t21/s0}$ . The closest integer is then $\var{b0}$ . Note that this solution uses the 3dp value of $B$ ; the answer using $\var{bb11}$ would give a slightly different long decimal value of $b$ , but the integer value would be the same.

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Taxonomy: mathcentre

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Newcastle University Mathematics and Statistics 9 years, 4 months ago

Error in variable testing condition

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Generated value: `number`

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Apply the sine rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`