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Trigonometry, SMB-061

Find the value of \(\theta\), correct to the nearest degree.   (3 marks)

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\(\theta=154^{\circ}\)

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\(\dfrac{\sin \theta}{21}\) \(= \dfrac{\sin 12^{\circ}}{10} \)  
\(\sin \theta\) \(= \dfrac{21 \times \sin 12^{\circ}}{10}\)  
\(\theta\) \(=\sin^{-1} (0.4366) \)  
  \(=25.88…^{\circ} \)  
  \(=26^{\circ}\ \text{(nearest degree)} \)  

 
\(\therefore\ \theta = 180-26=154^{\circ}\ \ \text{(angle is obtuse)}\)

Trigonometry, SMB-060

Find the value of \(\alpha\), correct to the nearest degree.   (2 marks)

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\(\alpha=39^{\circ}\)

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\(\dfrac{\sin \alpha}{9}\) \(= \dfrac{\sin 79^{\circ}}{14} \)  
\(\sin \alpha\) \(= \dfrac{9 \times \sin 79^{\circ}}{14}\)  
\(\alpha\) \(=\sin^{-1} (0.6310) \)  
  \(=39.12…^{\circ} \)  
  \(=39^{\circ}\ \text{(nearest degree)} \)  

Trigonometry, SMB-059

Find the value of \(\theta\), correct to the nearest degree.   (2 marks)

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\(\theta=50^{\circ}\)

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\(\dfrac{\sin \theta}{7.5}\) \(= \dfrac{\sin 66^{\circ}}{8.9} \)  
\(\sin \theta\) \(= \dfrac{7.5 \times \sin 66^{\circ}}{8.9}\)  
\(\theta\) \(=\sin^{-1} (0.7698) \)  
  \(=50.33…^{\circ} \)  
  \(=50^{\circ}\ \text{(nearest degree)} \)  

Trigonometry, SMB-058

Find the value of \(\theta\), correct to the nearest degree.   (2 marks)

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\(\theta=53^{\circ}\)

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\(\dfrac{\sin \theta}{6}\) \(= \dfrac{\sin 37^{\circ}}{4.5} \)  
\(\sin \theta\) \(= \dfrac{6 \times \sin 37^{\circ}}{4.5}\)  
\(\theta\) \(=\sin^{-1} (0.8024) \)  
  \(=53.361…^{\circ} \)  
  \(=53^{\circ}\ \text{(nearest degree)} \)  

Trigonometry, SMB-057

Find \(\alpha\), to the nearest degree, such that

\(\dfrac{\sin \alpha}{8} = \dfrac{\sin 60^{\circ}}{11} \)   (2 marks)

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\(\alpha=39^{\circ}\)

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\(\dfrac{\sin \alpha}{8}\) \(= \dfrac{\sin 60^{\circ}}{11} \)  
\(\sin \alpha\) \(= \dfrac{8 \times \sin 60^{\circ}}{11}\)  
\(\alpha\) \(=\sin^{-1} (0.6298) \)  
  \(=39^{\circ}\ \text{(nearest degree)} \)  

Trigonometry, SMB-056

Find \(\theta\), to the nearest degree, such that

\(\dfrac{12}{\sin \theta} = \dfrac{15}{\sin 26^{\circ}} \)   (2 marks)

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\(\theta=21^{\circ}\)

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\(\dfrac{12}{\sin \theta}\) \(= \dfrac{15}{\sin 26^{\circ}} \)  
\(\dfrac{\sin \theta}{12}\) \(= \dfrac{\sin 26^{\circ}}{15} \)  
\(\sin \theta\) \(= \dfrac{12 \times \sin 26^{\circ}}{15}\)  
\(\theta\) \(=\sin^{-1} (0.3507) \)  
  \(=21^{\circ}\ \text{(nearest degree)} \)  

Trigonometry, SMB-055

Find the value of \(x\), correct to 1 decimal place.   (2 marks)
 

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\(x=9.9\ \text{cm}\)

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\(\text{Angle needed}\ = 180-(115+34) = 31^{\circ}\)

\(\dfrac{x}{\sin31^{\circ}}\) \(=\dfrac{17.5}{\sin 115^{\circ}}\)  
\(x\) \(=\dfrac{17.5 \times \sin 31^{\circ}}{\sin 115^{\circ}}\)  
  \(=9.94…\)  
  \(=9.9\ \text{cm (to 1 d.p.)}\)  

Trigonometry, SMB-054

Find the value of \(x\), correct to 1 decimal place.   (2 marks)

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\(x=7.2\ \text{cm}\)

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\(\text{Angle needed}\ = 180-(81+43) = 56^{\circ}\)

\(\dfrac{x}{\sin56^{\circ}}\) \(=\dfrac{8.6}{\sin 81^{\circ}}\)  
\(x\) \(=\dfrac{8.6 \times \sin 56^{\circ}}{\sin 81^{\circ}}\)  
  \(=7.218…\)  
  \(=7.2\ \text{cm (to 1 d.p.)}\)  

Trigonometry, SMB-053

Find the value of \(x\), correct to 1 decimal place.  (2 marks)

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\(x=8.0\ \text{cm}\)

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\(\dfrac{x}{\sin42^{\circ}}\) \(=\dfrac{12}{\sin 86^{\circ}}\)  
\(x\) \(=\dfrac{12 \times \sin 42^{\circ}}{\sin 86^{\circ}}\)  
  \(=8.049…\)  
  \(=8.0\ \text{cm (to 1 d.p.)}\)  

Trigonometry, SMB-052

Find the value of \(x\), correct to 1 decimal place.  (2 marks)

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\(x=13.9\ \text{m}\)

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\(\dfrac{x}{\sin43^{\circ}}\) \(=\dfrac{20}{\sin 78^{\circ}}\)  
\(x\) \(=\dfrac{20 \times \sin 43^{\circ}}{\sin 78^{\circ}}\)  
  \(=13.944…\)  
  \(=13.9\ \text{cm (to 1 d.p.)}\)  

Trigonometry, SMC-051

Solve for \(b\), giving your answer correct to 1 decimal place.

\(\dfrac{b}{\sin 22^{\circ}} =  \dfrac{17}{\sin 67^{\circ}}\)  (2 marks)

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\(b=6.9\)

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\(\dfrac{b}{\sin 22^{\circ}}\) \(= \dfrac{17}{\sin 67^{\circ}}\) 
\(b\) \(=\dfrac{17 \times \sin 22^{\circ}}{\sin 67^{\circ}}\)
  \(=6.9\ \text{(to 1 d.p.)}\)

Trigonometry, SMB-050

Solve for \(a\), giving your answer correct to 1 decimal place.

\(\dfrac{6}{\sin 53^{\circ}} =  \dfrac{a}{\sin 27^{\circ}}\)  (2 marks)

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\(a=3.4\)

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\(\dfrac{6}{\sin 53^{\circ}}\) \(= \dfrac{a}{\sin 27^{\circ}}\) 
\(a\) \(=\dfrac{6 \times \sin 27^{\circ}}{\sin 53^{\circ}}\)
  \(=3.4\ \text{(to 1 d.p.)}\)

Measurement, STD2 M6 2022 HSC 26

The diagram shows two right-angled triangles, `ABC` and `ABD`,

where `AC=35 \ text{cm},BD=93 \ text{cm}, /_ACB=41^(@)` and `/_ADB=theta`.
 
     

Calculate the size of angle `theta`, to the nearest minute.  (4 marks)

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`19^@6^{′}`

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`text{In}\ Delta ABC:`

`cos 41^@` `=35/(BC)`  
`BC` `=35/(cos 41^@)`  
  `=46.375…`  

 
`angle BCD = 180-41=139^@`
 

`text{Using sine rule in}\ Delta BCD:`

`sin theta/(46.375)` `=sin139^@/93`  
`sin theta` `=(sin 139^@ xx 46.375)/93`  
`:.theta` `=sin^(-1)((sin 139^@ xx 46.375)/93)`  
  `=19.09…`  
  `=19^@6^{′}\ \ text{(nearest minute)}`  

♦ Mean mark 50%.

Measurement, STD2 M6 2005 HSC 5 MC

Which formula should be used to calculate the distance between Toby and Frankie?

  1. `a/(sin A) = b/(sin B)`
  2. `c^2 = a^2 + b^2`
  3. `A = 1/2 ab\ sinC`
  4. `c^2 = a^2 + b^2 − 2ab\ cosC`
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`A`

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`text(The triangle is not a right-angled triangle,)`

`:.\ text(Not)\ B`

`text(Given the information on the diagram provides)`

`text(2 angles and 1 side, the sine rule will work best.)`

`a/sinA = b/sinB`

`=> A`

Measurement, STD2 M6 2010 HSC 9 MC

Three towns `P`, `Q`  and `R` are marked on the diagram.

The distance from `R` to `P` is 76 km.  `angle RQP=26^circ`  and  `angle RPQ=46^@.`
 

 

  What is the distance from  `P`  to  `Q`  to the nearest kilometre?

  1. `100\ text(km)`
  2. `125\ text(km)`
  3. `165\ text(km)`
  4. `182\ text(km)`
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`C`

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`angle QRP` `=180-(26+46)     (180^circ\ text(in) \ Delta)`
  `=108^circ`

 

`text{Using sine rule}`

`(PQ)/sin108^circ` `=76/sin26^circ`
`PQ` `=(76xxsin108^circ)/sin26^circ`
  `=164.88\ text(km)`

`=>  C`