Apply the cosine 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 cosine rule	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Robert's copy of Apply the cosine rule	draft	Robert Pullan	18/07/2016 23:54
Robert's copy of Apply the cosine rule	draft	Robert Pullan	19/07/2016 00:26
Katy's copy of cosine rule	draft	Katy Dobson	08/08/2018 10:09
cormac's copy of Apply the cosine rule	draft	cormac breen	24/10/2018 17:08
Apply the cosine rule (3 sides given, degrees)	Ready to use	Andrew Chuter	06/04/2020 12:08

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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 three side lengths, determine the three angles using the Cosine Rule. Write down the angles (in radians) as decimals to 4dp. [Before submitting answers, you can check that the sum of the three angles is $\pi$ .]

Also calculate the area of the triangle, giving your answer as a decimal to 3dp.

Give any introductory information the student needs.

			Name	Type	Generated Value

c4	integer	16
s3	number	0.9682418185
cc0	number	1.2199
temp2	number	-0.0003425016
temp1	number	16.0003425016
b0	integer	16
cc3	number	1.3181
b3	integer	9
u2	number	0.9390993563
q0	number	0.34375
q3	number	0.875
c0	integer	16
area	number	82.636
cc5	number	1.318
s2	number	0.6457460822
s0	number	0.6455933588
cc1	number	1.2198926536
u0	number	0.939064987
u3	number	0.9682418185
cc2	number	1.220
aa5	number	1.318
aa4	number	1.3180926536
aa1	number	0.7017926536
aa0	number	0.7018
aa3	number	1.3181
aa2	number	0.702
x2	number	377
c31	integer	16
c32	integer	1
a0	integer	11
a3	integer	18
bb0	number	1.2199
s5	number	0.9682168121
c3	integer	18
c2	integer	19
c1	integer	12
x1	number	135
c02	integer	1
x4	number	243
x5	number	405
p3	number	0.25
p0	number	0.763671875
r0	number	0.34375
r3	number	0.25
bb3	number	0.5054
t5	number	0.4838074403
t2	number	0.9390993563
t3	number	0.4841574714
t0	number	0.939064987
u5	number	0.9682168121
c5	integer	20
cc4	number	1.3180926536
c01	integer	12
bb5	number	0.505
bb4	number	0.5053926536
check2	number	-0.0000073464
bb2	number	1.220
bb1	number	1.2198926536
check1	number	-0.0000073464

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
max(c31,c32)

JME function reference

Description

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Generated value: `integer`

← Depends on:

This variable doesn't seem to be used anywhere.

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 Number entry
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1. Step Information only

Gap-fill

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

$a=\var{a0}$ , $b=\var{b0}$ , $c=\var{c0}$

Angle $A=$ Gap 0.

Angle $B=$ Gap 1.

Angle $C=$ Gap 2.

Advice

(a) Use the Cosine Rule to find $\cos A$ : $\cos A =\dfrac{b^2+c^2-a^2}{2bc}$ . Therefore

$\cos A =\dfrac{\var{b0}^2+\var{c0}^2-\var{a0}^2}{2 \times \var{b0} \times \var{c0}}=\dfrac{\var{b0^2+c0^2-a0^2}}{\var{2 *b0*c0}}$

$=\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)}$

and so $A=\cos^{-1}(\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)})=\var{aa0}$ .

Similarly $\cos B =\dfrac{a^2+c^2-b^2}{2ac}$ and $\cos C =\dfrac{a^2+b^2-c^2}{2ab}$ . So

$\cos B =\dfrac{\var{a0}^2+\var{c0}^2-\var{b0}^2}{2 \times \var{a0} \times \var{c0}}=\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)}$

and so $B=\cos^{-1}(\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)})=\var{bb0}$ .

$\cos C =\dfrac{\var{a0}^2+\var{b0}^2-\var{c0}^2}{2 \times \var{a0} \times \var{b0}}=\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)}$

and so $C=\cos^{-1}(\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)})=\var{cc0}$ .

(b) We can use any of the formulae $\dfrac{1}{2}ac \sin B$ , $\dfrac{1}{2}bc \sin A$ or $\dfrac{1}{2}ab \sin C$ for the area. For example

$\dfrac{1}{2}bc \sin A = \dfrac{1}{2} \times \var{b0} \times \var{c0} \times \sin \var{aa0}$

$=\dfrac{1}{2} \times \var{b0 * c0} \times \var{sin(aa0)}=\var{Area}$

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Generated value: `integer`

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Apply the cosine 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

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`