Apply the sine rule

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 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	0.3615926536
cc1	number	2.5815926536
cc0	number	2.5815
b0	integer	18
cc2	number	2.582
check	number	0.0000926536
q0	number	0.9351851852
s2	number	0.1967088016
s0	number	0.1967088016
u1	number	0.5311137643
u0	number	0.5312646969
u2	number	0.5308410275
aa1	number	0.1980926536
aa0	number	0.1980
aa2	number	0.198
cc01	number	2.582
bb21	number	0.362
a0	integer	10
c0	integer	27
p0	number	0.9804526749
r0	number	-0.8472222222
t2	number	0.3541453211
t0	number	0.3541453211
t21	number	0.3541453211
c01	integer	22
c02	integer	27
bb2	number	0.362
bb1	number	0.3620926536
bb0	number	0.3620

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Value

xxxxxxxxxx
 
pi-aa0-cc01

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Description

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Generated value: `number`

0.3615926536

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Parts

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1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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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, 3 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, 3 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`