Sine rule and cosine rule Problem set B One question at a time

In triangle DEF, angle D = 44°, angle E = 63° and EF = 9.6 cm. Find the length of FD.

• EF is opposite D, and FD is opposite E
• \(\frac{x}{\sin 63°}=\frac{9.6}{\sin 44°}\)
• \(x=\frac{9.6\times\sin 63°}{\sin 44°}\), so from the calculator FD = 12.3 cm

In triangle XYZ, angle X = 58°, YZ = 12.3 cm and ZX = 9.4 cm. Find angle Y, which is acute.

• YZ is opposite X, and ZX is opposite Y
• \(\frac{\sin\theta}{9.4}=\frac{\sin 58°}{12.3}\)
• \(\sin\theta=\frac{9.4\times\sin 58°}{12.3}\)
• From the calculator: θ = sin⁻¹(0.6481) = 40.4°

In triangle KLM, angle L = 49°, MK = 15.8 cm and KL = 12.2 cm.

1. (a) Find angle M, which is acute.

   • KL is opposite M, and MK is opposite L
   • \(\frac{\sin M}{12.2}=\frac{\sin 49°}{15.8}\)
   • From the calculator: M = sin⁻¹(0.5828) = 35.6°

2. (b) Find angle K.

   • Angles in a triangle add to 180°
   • K = 180° − 49° − 35.6° = 95.4°

In triangle DEF, FD = 8.6 cm, DE = 5.3 cm and angle D = 125°. Find the length of EF.

• Two sides and the angle between them: use the cosine rule
• \(x^2=8.6^2+5.3^2-2\times8.6\times5.3\times\cos 125°\)
• \(x^2=154.3\)
• x = 12.4 cm

In triangle PQR, QR = 6.9 cm, RP = 9.3 cm and PQ = 12.0 cm. Find the largest angle.

• The largest angle is opposite the longest side, PQ: it is angle R
• \(\cos\theta=\frac{6.9^2+9.3^2-12.0^2}{2\times6.9\times9.3}\) = −0.07714
• θ = cos⁻¹(−0.07714) = 94.4°

The diagram shows triangle XYZ. The perpendicular from Z meets XY at S, and ZS = h. The side opposite X is x and the side opposite Y is y. Show that \(\frac{x}{\sin X}=\frac{y}{\sin Y}\).

• In triangle XSZ: \(\sin X=\frac{h}{y}\), so \(h=y\sin X\)
• In triangle YSZ: \(\sin Y=\frac{h}{x}\), so \(h=x\sin Y\)
• Both equal h: \(y\sin X=x\sin Y\)
• Divide both sides by \(\sin X\sin Y\): \(\frac{x}{\sin X}=\frac{y}{\sin Y}\)

Two lighthouses P and Q stand 250 m apart on a straight pier. A boat R is out at sea. Angle RPQ = 54° and angle RQP = 78°. Find the distance PR.

• Angle R = 180° − 54° − 78° = 48°
• Angle R is opposite the known side PQ = 250 m
• \(\frac{x}{\sin 78°}=\frac{250}{\sin 48°}\)
• \(x=\frac{250\times\sin 78°}{\sin 48°}\) = 329 m

Two hikers leave a camp C. One walks 6.4 km to a hut H and the other walks 9.1 km to a lake K. The angle HCK is 105°. Find the distance HK.

• The two sides and the angle between them are known: cosine rule
• \(x^2=9.1^2+6.4^2-2\times9.1\times6.4\times\cos 105°\)
• \(x^2=153.9\)
• x = 12.4 km

A triangular park has sides of 64 m, 96 m and 125 m. Find the largest angle of the park.

• The largest angle is opposite the longest side
• \(\cos\theta=\frac{64^2+96^2-125^2}{2\times64\times96}\) = −0.1882
• θ = cos⁻¹(−0.1882) = 101°

In triangle DEF, DE = 11.6 cm, DF = 15.2 cm and angle D = 37°.

1. (a) Find EF.

   • \(a^2=15.2^2+11.6^2-2\times15.2\times11.6\times\cos 37°\) = 83.97
   • EF = 9.16 cm

2. (b) Find angle EFD.

   • Sine rule with the side just found: \(\frac{\sin F}{11.6}=\frac{\sin 37°}{9.163}\)
   • F = sin⁻¹(0.7618) = 49.6°
   • The side opposite F (11.6 cm) is shorter than 15.2 cm, so F is acute

A flower bed PQRS has sides PQ = 7.5 m, QR = 5.8 m, RS = 8.3 m and SP = 6.6 m. Angle PQR = 75°.

1. (a) Find the length of PR.

   • \(PR^2=7.5^2+5.8^2-2\times7.5\times5.8\times\cos 75°\) = 67.37
   • PR = 8.21 m

2. (b) Find angle PSR.

   • \(\cos\theta=\frac{6.6^2+8.3^2-8.208^2}{2\times6.6\times8.3}\) = 0.4114
   • θ = cos⁻¹(0.4114) = 65.7°