Cosine Law Problems

Solve problems using the cosine law; a tutorial with detailed solutions and exercises with answers.

Problem 1

• Let us use the figure below and set
  
  $a=10\text{cm}$, $b=7\text{cm}$, and $c=5\text{cm}.$
  
  
• We now use the cosine law to find the largest angle $A$:
  $$a^2=b^2+c^2-2bc\cos (A)$$
• Substitute $a$, $b$, and $c$ by their values and solve for $\cos (A)$:
  $$\cos (A)=\frac{b^2+c^2-a^2}{2bc}$$ $$\cos (A)=\frac{7^2+5^2-{10}^2}{2\cdot 7\cdot 5}$$ $$\cos (A)=\frac{49+25-100}{70}=\frac{-13}{35}$$
• Use a calculator to find angle $A$ and round to 1 decimal place:
  $$A=\arccos \left(\frac{-13}{35}\right)\approx {111.8}^{\circ }$$
• We may use the sine law to find angle $B$:
  $$\frac{a}{\sin (A)}=\frac{b}{\sin (B)}$$
• Solve for $\sin (B)$:
  $$\sin (B)=\frac{b}{a}\sin (A)=\frac{7}{10}\sin ({111.8}^{\circ })$$
• Use a calculator to find $B$ and round to 1 decimal place:
  $$B\approx {40.5}^{\circ }$$
• Use the fact that the sum of all angles in a triangle is equal to ${180}^{\circ }$ to find angle $C$:
  $$C={180}^{\circ }-({40.5}^{\circ }+{111.8}^{\circ })\approx {27.7}^{\circ }$$

Problem 2

An aircraft tracking station determines the distance from a common point $O$ to each aircraft and the angle between the aircrafts. If the angle $O$ between the two aircrafts is ${49}^{\circ }$ and the distances from point $O$ to the two aircrafts are 50 km and 72 km, find the distance $d$ between the two aircrafts (round answers to 1 decimal place).

• A diagram to the above problem is shown below:
  
  
• The cosine law can be used as follows:
  $$d^2={72}^2+{50}^2-2(72)(50)\cos ({49}^{\circ })$$
• Solve for $d$:
  $$d=\sqrt{{72}^2+{50}^2-2(72)(50)\cos ({49}^{\circ })}\approx 54.4\text{km}$$

Exercises

1. A triangle has sides equal to 4 m, 11 m, and 8 m. Find its angles (round answers to 1 decimal place).

2. A ship leaves port at 1 pm traveling north at the speed of 30 miles/hour. At 3 pm, the ship adjusts its course 20 degrees eastward. How far is the ship from the port at 4 pm? (round to the nearest unit).

Solutions to Above Exercises

More References to Cosine and Sine Laws and Geometry