Review the law of sines and the law of cosines, and use them to solve problems with any triangle.

Law of sines

start fraction, a, divided by, sine, left parenthesis, alpha, right parenthesis, end fraction, equals, start fraction, b, divided by, sine, left parenthesis, beta, right parenthesis, end fraction, equals, start fraction, c, divided by, sine, left parenthesis, gamma, right parenthesis, end fraction

Law of cosines

c, start superscript, 2, end superscript, equals, a, start superscript, 2, end superscript, plus, b, start superscript, 2, end superscript, minus, 2, a, b, cosine, left parenthesis, gamma, right parenthesis
Want to learn more about the law of sines? Check out this video.
Want to learn more about the law of cosines? Check out this video.

Practice set 1: Solving triangles using the law of sines

This law is useful for finding a missing angle when given an angle and two sides, or for finding a missing side when given two angles and one side.

Example 1: Finding a missing side

Let's find A, C in the following triangle:
According to the law of sines, start fraction, A, B, divided by, sine, left parenthesis, angle, C, right parenthesis, end fraction, equals, start fraction, A, C, divided by, sine, left parenthesis, angle, B, right parenthesis, end fraction. Now we can plug the values and solve:
ABsin(C)=ACsin(B)5sin(33)=ACsin(67)5sin(67)sin(33)=AC8.45AC\begin{aligned} \dfrac{AB}{\sin(\angle C)}&=\dfrac{AC}{\sin(\angle B)} \\\\ \dfrac{5}{\sin(33^\circ)}&=\dfrac{AC}{\sin(67^\circ)}\\\\ \dfrac{5\sin(67^\circ)}{\sin(33^\circ)}&=AC \\\\ 8.45&\approx AC \end{aligned}

Example 2: Finding a missing angle

Let's find m, angle, A in the following triangle:
According to the law of sines, start fraction, B, C, divided by, sine, left parenthesis, angle, A, right parenthesis, end fraction, equals, start fraction, A, B, divided by, sine, left parenthesis, angle, C, right parenthesis, end fraction. Now we can plug the values and solve:
BCsin(A)=ABsin(C)11sin(A)=5sin(25)11sin(25)=5sin(A)11sin(25)5=sin(A)\begin{aligned} \dfrac{BC}{\sin(\angle A)}&=\dfrac{AB}{\sin(\angle C)} \\\\ \dfrac{11}{\sin(\angle A)}&=\dfrac{5}{\sin(25^\circ)} \\\\ 11\sin(25^\circ)&=5\sin(\angle A) \\\\ \dfrac{11\sin(25^\circ)}{5}&=\sin(\angle A) \end{aligned}
Evaluating using the calculator and rounding:
m, angle, A, equals, sine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, 11, sine, left parenthesis, 25, degree, right parenthesis, divided by, 5, end fraction, right parenthesis, approximately equals, 68, point, 4, degree
Remember that if the missing angle is obtuse, we need to take 180, degree and subtract what we got from the calculator.
Problem 1.1
B, C, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Round to the nearest tenth.

Want to try more problems like this? Check out this exercise.

Practice set 2: Solving triangles using the law of cosines

This law is mostly useful for finding an angle measure when given all side lengths. It's also useful for finding a missing side when given the other sides and one angle measure.

Example 1: Finding an angle

Let's find m, angle, B in the following triangle:
According to the law of cosines:
left parenthesis, A, C, right parenthesis, start superscript, 2, end superscript, equals, left parenthesis, A, B, right parenthesis, start superscript, 2, end superscript, plus, left parenthesis, B, C, right parenthesis, start superscript, 2, end superscript, minus, 2, left parenthesis, A, B, right parenthesis, left parenthesis, B, C, right parenthesis, cosine, left parenthesis, angle, B, right parenthesis
Now we can plug the values and solve:
(5)2=(10)2+(6)22(10)(6)cos(B)25=100+36120cos(B)120cos(B)=111cos(B)=111120\begin{aligned} (5)^2&=(10)^2+(6)^2-2(10)(6)\cos(\angle B) \\\\ 25&=100+36-120\cos(\angle B) \\\\ 120\cos(\angle B)&=111 \\\\ \cos(\angle B)&=\dfrac{111}{120} \end{aligned}
Evaluating using the calculator and rounding:
m, angle, B, equals, cosine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, 111, divided by, 120, end fraction, right parenthesis, approximately equals, 22, point, 33, degree

Example 2: Finding a missing side

Let's find A, B in the following triangle:
According to the law of cosines:
left parenthesis, A, B, right parenthesis, start superscript, 2, end superscript, equals, left parenthesis, A, C, right parenthesis, start superscript, 2, end superscript, plus, left parenthesis, B, C, right parenthesis, start superscript, 2, end superscript, minus, 2, left parenthesis, A, C, right parenthesis, left parenthesis, B, C, right parenthesis, cosine, left parenthesis, angle, C, right parenthesis
Now we can plug the values and solve:
(AB)2=(5)2+(16)22(5)(16)cos(61)(AB)2=25+256160cos(61)AB=281160cos(61)AB14.3\begin{aligned} (AB)^2&=(5)^2+(16)^2-2(5)(16)\cos(61^\circ) \\\\ (AB)^2&=25+256-160\cos(61^\circ) \\\\ AB&=\sqrt{281-160\cos(61^\circ)} \\\\ AB&\approx 14.3 \end{aligned}
Problem 2.1
m, angle, A, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
degree
Round to the nearest degree.

Want to try more problems like this? Check out this exercise.

Practice set 3: General triangle word problems

Problem 3.1
"Only one remains." Ryan signals to his brother from his hiding place.
Matt nods in acknowledgement, spotting the last evil robot.
"34 degrees." Matt signals back, informing Ryan of the angle he observed between Ryan and the robot.
Ryan records this value on his diagram (shown below) and performs a calculation. Calibrating his laser cannon to the correct distance, he stands, aims, and fires.
To what distance did Ryan calibrate his laser cannon?
Do not round during your calculations. Round your final answer to the nearest meter.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
space, m

Want to try more problems like this? Check out this exercise.