Law of Sines
What is Sine Law?
Sine is a trigonometric ratio and this law describes the relationship between the sides and their corresponding angles within a non-right angle triangle. A non-right angle triangle is also known as oblique.
It states that the ratio of a side and its corresponding angle is the same for all the sides and angles within a particular triangle.
Equation of Sine Law
Where lowercase alphabets represent the sides (length) and the upper case alphabets represent angles.
Both equations are the same and can be used interchangeably.
When to use Sine Law?
This law is used when the triangle in consideration is a non-right angled triangle i.e oblique. Moreover, it can be used to find the missing length or angle when the other three are known i.e. two angles and one of the corresponding sides or two sides and one of the corresponding angles.
Example of Sine Law
Example 1: Find angle A and side c.
Explanation:
Finding angle A: Side a, angle B and side b are known. We plug them in the formula. 5.9 was divided on the right side so when it is taken to the left side it is multiplied. Calculate the sin of the angle using your calculator then multiply it with 5.9 and finally divide it with 8.4. Do not round off just yet, make sure you keep your answer in 4-5 decimal places. Take the inverse sin of the number you got to get the final answer.
Finding side c: In order to find side c angle C needs to be known. As we know the sum of angles in a triangle is equal to 180 so we can subtract the sum of angle A and B from 180 to get angle C.
Then plug the values of Angle C, Angle B and side b in formula. Cross multiply and then divide sin78 on the right side as it was multiplied on the left. Then again using your calculator, calculate the sin of 59, multiply it with 8.4 and divide the resulting answer with sin of 78.
Now it's your turn to try.
Practice problems for the Sine Law
1. In 
, 
C = 75
, b = 5cm and c = 19cm. Solve the triangle. Round to the nearest whole number.
2. In , A = 54, a = 2.5m and c = 1.6m . Solve the triangle. Round to the nearest whole number.
3. In , B = 48, a = 28cm and A = 35. Solve the triangle. Round to the nearest whole number.
Take home points for Sine Law
• Make sure the triangle is not right-angles otherwise you can use the simple trigonometric ratios (SOH CAH TOA)
• Make sure the triangle is not right-angles otherwise you can use the simple trigonometric ratios (SOH CAH TOA)
• Take sin of the angle first and then multiply or divide
• Make sure you do not round off your answers when you take sin of the angle. Only round of in the final answer according to what is asked in the question (e.g. nearest degree or tenth of a cm)
• Draw the triangle to visualize the question whenever not provided with a triangle. Solve the triangle means solving for all unknown sides and angles.
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