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, 3 months ago

Created this.

There is only one version of this question that you have access to.

There are 4 other versions that do you not have access to.

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 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

s3	number	0.774421371
cc0	number	2.2718
b0	integer	21
cc3	number	1.7404
b3	integer	14
cc2	number	2.272
check	number	-0.0000073464
q0	number	0.9114219114
q3	number	0.8701298701
area	number	176.531
cc5	number	1.740
s2	number	0.4313602756
s0	number	0.4310896023
cc1	number	2.2717926536
u0	number	0.7641952181
u3	number	0.985651741
u2	number	0.7640662058
aa5	number	0.886
aa4	number	0.8858926536
aa1	number	0.4456926536
aa0	number	0.4457
aa3	number	0.8858
aa2	number	0.446
c31	integer	28
c32	integer	34
a0	integer	22
a3	integer	22
s5	number	0.7745478896
c3	integer	28
c0	integer	39
c02	integer	40
p3	number	0.6326530612
p0	number	0.9023199023
r0	number	-0.645021645
r3	number	-0.1688311688
bb3	number	0.5153
t5	number	0.492534849
t2	number	0.411409537
t3	number	0.4927959147
t0	number	0.41150068
u5	number	0.9857191788
cc4	number	1.7404926536
c01	integer	32
bb5	number	0.515
bb4	number	0.5153926536
check2	number	0.0000926536
bb2	number	0.424
bb1	number	0.4240926536
bb0	number	0.4241

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
sin(AA3)

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: `number`

0.774421371

← 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

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

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, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

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, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`