The law of sines is a useful rule showing a relationship between an angle of a triangle and the length of the side opposite of the angle.
The law is expressed by the formula
The sine of the angle divided by the length of the opposite side is the same for every angle and its opposing side of the triangle.
Law of Sines – How does it work?
It is easy to show how this law works. First, let’s take the triangle from above and drop a vertical line to the side marked c.
This cuts the triangle into two right triangles which share a common side marked h.
The sine of an angle in a right triangle is the ratio of the length of the side opposite of the angle to the length of the hypotenuse of the right triangle. In other words:
Take the right triangle including the angle A. The length of the side opposite of A is h and the hypotenuse is equal to b.
Solve this for h and get
h = b sin A
Do the same thing for the right triangle including angle B. This time, the length of the side opposite of B is still h but the hypotenuse is equal to a.
Solve this for h and get
h = a sin B
Since both of these equations are equal to h, they are equal to each other.
b sin A = a sin B
We can rewrite this to get the same letters on the same side of the equation to get
You can repeat
Law of Sines Example Problem
Question: Use the law of sines to find the length of the side x.
Solution: The unknown side x is opposite the 46.5° angle and the side with length 7 is opposite the 39.4° angle. Plug these values into the Law of Sines equation.
Solve for x
7 sin(46.5°) = x sin(39.4°)
7 (0.725) = x (0.635)
5.078 = x (0.635)
x = 8
Answer: The unknown side is equal to 8.
Bonus: If you wanted to find the missing angle and length of the last side of the triangle, remember that all three angles of a triangle all add up to 180°.
180° = 46.5° + 39.4° + C
C = 94.1°
Use this angle in the law of sines the same way as above with either of the other angles and get a length of side c equal to 11.
Potential Issue of the Law of Sines
One potential problem to keep in mind using the law of sines is the possibility of two answers for an angle variable. This tends to appear when you are given two side values and an acute angle not between the two sides.
These two triangles are an example of this problem. The two sides are 100 and 75 in length and the 40° angle is not between these two sides.
Notice how the side with length 75 could swing to hit a second place along the bottom side. Both of these angles will give a valid answer using the law of sines.
Fortunately, these two angle solutions add up to 180°. This is because the triangle formed by the two 75 sides is an isosceles triangle (triangle with two equal sides). The angles between the sides and their shared side will also be equal to each other. This means the angle on the other side of the angle θ will be the same as angle φ. The two angles added together make a straight line, or 180°.
Law of Sines Example Problem 2
Question: What are the two possible angles of a triangle with sides of 100 and 75 with a 40° as marked in the triangles above?
Solution: Use the law of sines formula where the 75 length is opposite of 40°, and 100 is opposite of θ.
sin θ = 0.857
θ = 58.97°
θ + φ = 180°
φ = 180° – θ
φ = 180° – 58.97°
φ = 121.03°
Answer: The two possible angles for this triangle is 58.97° and 121.03°.
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