AI HL · 12.10
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Sine rule and cosine rule
12.10.2
Screenshot and video to come
Screenshot and video to come
Drag the corners to make any triangle. Tick the three measurements you are given, then choose what to find. The applet names the rule and writes the working.
Tick three measurements.
In triangle PQR, angle P = 48°, angle Q = 65° and QR = 11.4 cm. Find the length of RP.
Have a go first. Then open the working one line at a time.
QR is opposite P. PR is opposite Q.
Each side goes with the angle opposite it.
\(\frac{x}{\sin 65°}=\frac{11.4}{\sin 48°}\)
Put the unknown on top so you can multiply.
\(x=\frac{11.4\times\sin 65°}{\sin 48°}\)
Multiply both sides by sin 65°.
From the calculator: PR = 13.9 cm
The larger angle is opposite the longer side: 65° is larger than 48°, so PR is longer than QR.
In triangle ABC, angle A = 72°, BC = 12.4 cm and CA = 8.9 cm. Find angle B, which is acute.
Have a go first. Then open the working one line at a time.
BC = 12.4 is opposite A. AC = 8.9 is opposite B.
Use the pair you know in full: side 12.4 with angle 72°.
\(\frac{\sin B}{8.9}=\frac{\sin 72°}{12.4}\)
To find an angle, put the sines on top.
\(\sin B=\frac{8.9\times\sin 72°}{12.4}\) = 0.6826
From the calculator.
B = sin⁻¹(0.6826) = 43.0°
AC is shorter than BC, so B is smaller than 72°: the acute answer is right.
In triangle ABC, angle B = 58°, CA = 14.6 cm and AB = 11.3 cm. Find angle A.
Have a go first. Then open the working one line at a time.
\(\frac{\sin C}{11.3}=\frac{\sin 58°}{14.6}\)
AB = 11.3 is opposite C. AC = 14.6 is opposite B.
From the calculator: C = sin⁻¹(0.6564) = 41.0°
AB is shorter than AC, so C is acute.
A = 180° − 58° − 41.0° = 81.0°
Angles in a triangle add to 180°. Keep the unrounded C on the calculator.
In triangle ABC, CA = 7.3 cm, AB = 10.4 cm and angle A = 64°. Find the length of BC.
Have a go first. Then open the working one line at a time.
Two sides and the angle between them
The sine rule needs a side and its opposite angle. Here there is none: use the cosine rule.
\(a^2=7.3^2+10.4^2-2\times7.3\times10.4\times\cos 64°\)
The angle is the one opposite the side you want.
\(a^2=94.89\)
From the calculator.
BC = √94.89 = 9.74 cm
A length is the positive root.
In triangle ABC, AB = 9.1 cm, AC = 7.5 cm and BC = 11.2 cm. Find angle A.
Have a go first. Then open the working one line at a time.
Three sides: use the cosine rule rearranged for the angle
The angle is opposite the side that is on its own: BC = 11.2.
\(\cos A=\frac{b^2+c^2-a^2}{2bc}\)
This form saves rearranging each time.
\(\cos A=\frac{7.5^2+9.1^2-11.2^2}{2\times7.5\times9.1}\) = 0.09978
From the calculator.
A = cos⁻¹(0.09978) = 84.3°
This is the largest angle, as BC is the longest side.
In triangle ABC, the perpendicular from C meets AB at D and CD = h. Side b = AC = 7.0 cm and angle A = 40°. (a) Find h. (b) Write h in terms of a and sin B, and show that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Have a go first. Then open the working one line at a time.
\(\sin A=\frac{h}{b}\), so \(h=b\sin A\) = 7.0 × sin 40° = 4.50 cm
CD is opposite A and AC is the hypotenuse.
\(\sin B=\frac{h}{a}\), so \(h=a\sin B\)
CD is opposite B and BC is the hypotenuse.
\(b\sin A=a\sin B\)
Both equal h.
\(\frac{a}{\sin A}=\frac{b}{\sin B}\)
Divide both sides by sin A sin B. The same works for the third side c, using a perpendicular from A.