# 4 Tips for Solving Trigonometric Problems using Sine rule

The Sine Rule is suitable for evaluating various properties of triangles. It's not necessary for the triangle to be a right-angled triangle for the application of the sine rule. This rule can also be applied for evaluating properties of scalene triangles. This article will enable you to get some ideas of the Sine rule application in solving various trigonometric problems, vector problems in higher mathematics etc. through 4 simple steps.

## What does the Sine Rule or Law state?

The rule or law states that “ratio of length of one side of a triangle to the sine of its opposite angle is same for all 3 sides of the triangle.”

Take this triangle in the following diagram as an example.

Fig. 1

Image courtesy- bit.ly/1UWm1hd

According to the above diagram and the Sine rule, the equation should go like this-

a/sin A = b/sin B = c/sin C

Let's go through the steps for solving trigonometric problems using Sine rule. An example is also provided for an easy explanation.

## Tips

• Determine the Necessary Values required for Calculation and note them down

Look at the Sine Rule equation above and find out the values that are required for your calculation from the problem itself. So what are the values required for your calculation? You have to know the values of at least “one angle and 2 sides” or “two angles and one side”. A, B, C are the angles and a, b, c are the sides. The values required for solving this sum should be provided in the problem itself. Write them down on your sheet for an easy reference. This will be immensely helpful to you because you have actually determined what you “know” and what you “do not know” that needs to be solved. Writing down those values in a definite order will be convenient to you because you don't have to refer to the question again and again while substituting those values in your equation and make mistakes in the process.

It is advisable to convert the angles in radians for easy calculation. This is also the proper procedure for a more accurate answer. If the angle given is in degrees, convert it into radians. An example is shown below:

Let's say the angle given is 30°. There's a simple process to calculate it in radians. Follow this procedure for converting “any” angle in degrees to radians.

π/x= given angle

or, π/x=30°

(Now, π= 180°)

or, 180° /30° =x

or, x=6.

Therefore, the angle in radians is π/6.

• Find out the value of the third unknown angle if it's needed

For calculating the third unknown angle, you'll have to know the other two angles that are required for calculation. The third angle generally doesn't need to be evaluated. If the third angle consists of an unknown angle which is opposite a side that needs to be evaluated, you have to find out the value of the third angle. The process is simple subtraction of the sum of two known angles from 180°. The result will give you the value of the third angle.

• Choose the right combination of angles and sides required for the solution

For incorporating the equation into your solution, you need to choose the right combination of angles and sides from the problem. Say for example, you know the values of a, c and A in the Fig. 1 depicted above. You have to calculate the value of C. You need to take the following combination of angles and sides to form an equation-

• Substitute the values that are known to you into the equation and solve the Problem

Put the correct values into their respective places in the equation and start applying basic algebraic cum trigonometric operations to solve the equation. Do pay special attention to the substitution of values and algebraic operations because a single mistake in performing such simple operations will alter your entire result. The values of sine of unknown angles should be provided in the problem itself like sin 25°, sin 67° etc. It's expected that you know the values of the sine of certain angles like 0°, 30°, 45°, 60°, 90°. You can memorize these values from the following table.

 Trigonometric Angles Values sin 0° 0 sin 30° 1/2 sin 45° 1/√2 sin 60° √3/2 sin 90° 1

You can easily determine the values of sine of unknown angles from your calculator. If you are not allowed a calculator, then you would be provided the values in the problem itself unless it involves the ones provided in the table above.

## Example

You are provided with the following problem shown in Fig. 2. You have to determine the value of AC. The sine rule application is best suited for this purpose.

Fig. 2

In the above figure, the angle which lies opposite to AC (the side which needs to be evaluated) is known. Hence, there's no need to calculate the third unknown angle which is ∠ACB.

Given, ∠ABC= 27°

∠BAC= 123°

BC   = 12 cm.

Solution,

Applying Sine Rule,

AC/ sin ∠ABC= BC/ sin ∠BAC

or, AC/ sin 27° = BC/ sin 123°,

or, AC/ sin 27° = 12/ sin 123°

or, AC/0.454 = 12/ 0.838

AC= (12 x 0.454)/ 0.838

or, AC= 6.5

You see that Sine rule problems are not that difficult. Once you get a firm grasp of this concept, the sums will be easy. Keep on practising these kind of sums for a better performance. Best of luck!

Image by: Kyle Sherrets www.flickr.com

### Sudipto Das

Sudipto writes technical and educational content periodically for wizert.com and backs it up with extensive research and relevant examples. He's an avid reader and a tech enthusiast at the same time with a little bit of “Arsenal Football Club” thrown in as well. He's got a B.Tech in Electronics and Instrumentation.