Apply the sine rule

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Created this.

Name	Status	Author	Last Modified
Apply the sine rule	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Katy's Apply the sine rule	draft	Katy Dobson	08/08/2018 10:08
Apply the sine rule (1 angle, 2 sides, degrees)	Ready to use	Andrew Chuter	08/04/2020 13:13

There are 9 other versions that do you not have access to.

Question statement

Suppose that $\Delta ABC$ is a triangle with all interior angles $< \dfrac{\pi}{2}$ (in other words, an acute 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 angle and two side lengths, use the Sine Rule to determine the other side length and two angles. Write down the side length as a whole number and the angles (in radians) as decimals to 3dp.

Give any introductory information the student needs.

			Name	Type	Generated Value

c4	integer	9
s3	number	0.6614121415
cc0	number	1.0989
temp2	number	0.0000097929
temp1	number	21.9999902071
b0	integer	22
cc3	number	0.9734
b3	integer	12
u2	number	0.8907533184
bb51	number	1.445
q0	number	0.4545454545
q3	number	0.125
c0	integer	22
aa4	number	0.7226926536
cc5	number	0.973
s2	number	0.8099107858
s0	number	0.8097347834
cc1	number	1.0989926536
u0	number	0.8907078652
u3	number	0.8268029602
cc2	number	1.099
aa5	number	0.723
cc21	number	1.099
aa1	number	0.9437926536
aa0	number	0.9437
aa3	number	0.7227
aa2	number	0.944
x2	number	884
c31	integer	9
c32	integer	1
bb01	number	1.099
t51	number	0.9920980708
a0	integer	20
a3	integer	8
bb0	number	1.0989
s4	number	0.6614423682
s5	number	0.6616371185
c3	integer	10
c2	integer	29
c1	integer	10
x1	number	84
bb41	number	1.4451926536
x4	number	80
x5	number	208
p3	number	0.75
p0	number	0.5867768595
r0	number	0.4545454545
r3	number	0.5625
bb3	number	1.4455
aa31	number	0.723
t5	number	0.9920980708
t2	number	0.8907533184
t3	number	0.9921606792
t0	number	0.8907078652
t1	number	0.8907082617
u5	number	0.8265778974
c02	integer	1
c5	integer	14
u21	number	0.8907533184
cc4	number	0.9733926536
cc11	number	1.0988926536
c01	integer	10
bb5	number	1.445
bb4	number	1.4454926536
check2	number	-0.0000073464
bb2	number	1.099
bb1	number	1.0989926536
check1	number	0.0000926536

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
max(c31,c32)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

← Depends on:

→ Used by:

Parts

Part a) Gap-fill
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
Part b) Gap-fill
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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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$A=\var{AA0}$ , $a=\var{a0}$ , $b=\var{b0}$

Angle $B=$ Gap 0.

Angle $C=$ Gap 1.

Side length $c=$ Gap 2.

Advice

a) We use the Sine Rule to find $B$ : $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$ . Thus $\sin B=\dfrac{b \sin A}{a}=\dfrac{\var{b0}* \var{s0}}{\var{a0}}=\var{b0*s0/a0}$ . To find $B$ we need to calculate $\sin^{-1} (\var{b0*s0/a0})$ , calculating the angle between $0$ and $\dfrac{\pi}{2}$ , so $B=\var{bb01}$ (to 3 decimal places).

Since $A+B+C=\pi$ , we calculate $C=\pi-A-B=\var{CC11}$ . To 3dp, this gives $\var{CC21}$ .

We use the Sine Rule to find $c$ : $\dfrac{a}{\sin A}=\dfrac{c}{\sin C}$ . Thus $c=\dfrac{a \sin C}{\sin A}=\dfrac{\var{a0}* \var{u21}}{\var{s0}}=\var{a0*u21/s0}$ . The closest integer is then $\var{c0}$ . Note that this solution uses the 3dp value of $C$ ; the answer using $\var{CC11}$ would give a slightly different long decimal value of $c$ , but the integer value would be the same.

b) We use the Sine Rule to find $A$ : $\dfrac{a}{\sin A}=\dfrac{c}{\sin C}$ . Thus $\sin A=\dfrac{a \sin C}{c}=\dfrac{\var{a3}* \var{u3}}{\var{c3}}=\var{a3*u3/c3}$ . To find $A$ we need to calculate $\sin^{-1} (\var{a3*u3/c3})$ , calculating the angle between $0$ and $\dfrac{\pi}{2}$ , so $A=\var{aa31}$ (to 3 decimal places).

Since $A+B+C=\pi$ , we calculate $B=\pi-A-C=\var{bb41}$ . To 3dp, this gives $\var{bb51}$ .

We use the Sine Rule to find $b$ : $\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ . Thus $b=\dfrac{c \sin B}{\sin C}=\dfrac{\var{c3}* \var{t51}}{\var{u3}}=\var{c3*t51/u3}$ . The closest integer is then $\var{b3}$ . Note that this solution uses the 3dp value of $B$ ; the answer using $\var{bb41}$ would give a slightly different long decimal value of $b$ , but the integer value would be the same.

Give a worked solution to the whole question.

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Taxonomy: mathcentre

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Taxonomy: Context

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

Gap-fill

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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, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`