Faded examples · Using the sine rule to find a side

Calculator allowed Calculator

Each example shows a little less than the one before – complete the faded steps yourself, using the same three steps every time: write the sine rule for the pair, substitute, then rearrange and evaluate to 1 d.p. Also answer the check question. Calculator in DEGREES.

①Example 1fully worked: read it through
In triangle $ABC$, $A=40^\circ$, $B=75^\circ$ and $a=8$ cm. Find $b$.Triangle ABC (not to scale)ABC8 cmb40°75°Not to scale
1Write the sine rule for the pair
$\dfrac{b}{\sin B}=\dfrac{a}{\sin A}$
2Substitute the known values
$\dfrac{b}{\sin75^\circ}=\dfrac{8}{\sin40^\circ}$
3Rearrange and evaluate (1 d.p.)
$b=\dfrac{8\sin75^\circ}{\sin40^\circ}=12.0$ cm
Check · Which side-angle pair lets us start?
A $a$ and $A$ – a side with its opposite angleB $b$ and $A$C $a$ and $B$D any two angles
②Example 2you finish the last 1 step
In triangle $ABC$, $A=35^\circ$, $B=80^\circ$ and $a=6$ cm. Find $b$.Triangle ABC (not to scale)ABC6 cmb35°80°Not to scale
1Write the sine rule for the pair
$\dfrac{b}{\sin80^\circ}=\dfrac{6}{\sin35^\circ}$
2Substitute the known values
$b=\dfrac{6\sin80^\circ}{\sin35^\circ}$
3Rearrange and evaluate (1 d.p.)
$b=10.3$ cm
Check · To make $b$ the subject you...
A add $\sin80^\circ$ to both sidesB multiply both sides by $\sin80^\circ$C divide by $6$D square both sides
③Example 3you finish the last 2 steps
In triangle $ABC$, $A=50^\circ$, $B=60^\circ$ and $a=10$ cm. Find $b$.Triangle ABC (not to scale)ABC10 cmb50°60°Not to scale
1Write the sine rule for the pair
$\dfrac{b}{\sin60^\circ}=\dfrac{10}{\sin50^\circ}$
2Substitute the known values
$b=\dfrac{10\sin60^\circ}{\sin50^\circ}$
3Rearrange and evaluate (1 d.p.)
$b=11.3$ cm
Check · Why is $b$ bigger than $a$ here?
A a calculation errorB angle $B$ is bigger than angle $A$, so its opposite side is longerC sides are always increasingD because $60>50$ has no effect
④Example 4your turn: every step
In triangle $ABC$, $A=48^\circ$, $B=72^\circ$ and $a=9$ cm. Find $b$.Triangle ABC (not to scale)ABC9 cmb48°72°Not to scale
1Write the sine rule for the pair
$\dfrac{b}{\sin72^\circ}=\dfrac{9}{\sin48^\circ}$
2Substitute the known values
$b=\dfrac{9\sin72^\circ}{\sin48^\circ}$
3Rearrange and evaluate (1 d.p.)
$b=11.5$ cm
Check · What must the calculator be set to?
A radiansB degreesC gradiansD it does not matter

Answers · Sine & Cosine Rule

Faded examples · Using the sine rule to find a side
① Example 1   $\dfrac{b}{\sin B}=\dfrac{a}{\sin A}$→$\dfrac{b}{\sin75^\circ}=\dfrac{8}{\sin40^\circ}$→$b=\dfrac{8\sin75^\circ}{\sin40^\circ}=12.0$ cm   $12.0\text{ cm}$
Check: A: $a$ and $A$ – a side with its opposite angle
② Example 2   $\dfrac{b}{\sin80^\circ}=\dfrac{6}{\sin35^\circ}$→$b=\dfrac{6\sin80^\circ}{\sin35^\circ}$→$b=10.3$ cm   $10.3\text{ cm}$
Check: B: multiply both sides by $\sin80^\circ$
③ Example 3   $\dfrac{b}{\sin60^\circ}=\dfrac{10}{\sin50^\circ}$→$b=\dfrac{10\sin60^\circ}{\sin50^\circ}$→$b=11.3$ cm   $11.3\text{ cm}$
Check: B: angle $B$ is bigger than angle $A$, so its opposite side is longer
④ Example 4   $\dfrac{b}{\sin72^\circ}=\dfrac{9}{\sin48^\circ}$→$b=\dfrac{9\sin72^\circ}{\sin48^\circ}$→$b=11.5$ cm   $11.5\text{ cm}$
Check: B: degrees
mathedup.co.uk · sheet VT0L