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

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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 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	0.3530228379
cc0	number	1.7669
b0	integer	17
cc3	number	1.8127
b3	integer	28
cc2	number	1.767
check	number	-0.0000073464
q0	number	0.853515625
q3	number	0.5669191919
area	number	200.092
cc5	number	1.813
s2	number	0.7359034262
s0	number	0.7356325327
cc1	number	1.7668926536
u0	number	0.9808332172
u3	number	0.970883707
u2	number	0.9808137273
aa5	number	0.361
aa4	number	0.3608926536
aa1	number	0.8265926536
aa0	number	0.8266
aa3	number	0.3608
aa2	number	0.827
c31	integer	32
c32	integer	38
a0	integer	24
a3	integer	12
s5	number	0.3532099538
c3	integer	33
c0	integer	32
c02	integer	39
p3	number	0.9356060606
p0	number	0.6773897059
r0	number	-0.1948529412
r3	number	-0.2395833333
bb3	number	0.9680
t5	number	0.8237534653
t2	number	0.5209811356
t3	number	0.8237534653
t0	number	0.5210664899
u5	number	0.9708117979
cc4	number	1.8127926536
c01	integer	31
bb5	number	0.968
bb4	number	0.9680926536
check2	number	0.0000926536
bb2	number	0.548
bb1	number	0.5480926536
bb0	number	0.5481

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.

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Generated value: `number`

0.3530228379

← 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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Gap-fill

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

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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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

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`