Apply the cosine rule

Error

There was an error loading the page.

Contributors

Newcastle University Mathematics and Statistics

History

○

Newcastle University Mathematics and Statistics 9 years, 4 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

There are 7 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 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	15
s3	number	0.8627078041
cc0	number	1.5598
temp2	number	0.0008150535
temp1	number	13.9991849465
b0	integer	14
cc3	number	1.5247
b3	integer	12
u2	number	0.9999417202
q0	number	0.6761133603
q3	number	0.8385167464
c0	integer	19
area	number	90.998
cc5	number	1.525
s2	number	0.683830756
s0	number	0.6841954908
cc1	number	1.5597926536
u0	number	0.999939541
u3	number	0.9989377524
cc2	number	1.560
aa5	number	1.041
aa4	number	1.0406926536
aa1	number	0.7534926536
aa0	number	0.7535
aa3	number	1.0406
aa2	number	0.753
x2	number	365
c31	integer	15
c32	integer	1
a0	integer	13
a3	integer	19
bb0	number	0.8283
s5	number	0.8629100162
c3	integer	22
c2	integer	19
c1	integer	6
x1	number	27
c02	integer	1
x4	number	217
x5	number	505
p3	number	0.5056818182
p0	number	0.7293233083
r0	number	0.010989011
r3	number	0.0460526316
bb3	number	0.5762
t5	number	0.5446737109
t2	number	0.7365801446
t3	number	0.5448414296
t0	number	0.7367830166
u5	number	0.9989515315
c5	integer	22
cc4	number	1.5247926536
c01	integer	6
bb5	number	0.576
bb4	number	0.5762926536
check2	number	0.0000926536
bb2	number	0.828
bb1	number	0.8282926536
check1	number	-0.0000073464

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:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
1. Step Information only
Part b) Gap-fill
1. Gap Number entry
  Add
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

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

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

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Information only
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Information only

This question is used in the following exams:

Loading...

Error

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: `integer`

Parts

Gap-fill

Loading...

Error

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, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`