How to Solve Trigonometric Problems Using the Sine Rule

The Sine rule is perfectly suitable for evaluating the different properties of a triangle irrespective of it being scalene, isosceles, or a right-angled triangle.

This article will throw light on ways you can use the Sine rule for solving trigonometric and vector problems in higher school mathematics. So brace yourself and let’s begin.

The Sine Rule Statement

According to the Sine rule, the ratio of the length of one side of a triangle to the sine of its opposite angle is same for all 3 sides of the triangle.

Here, have a look at the following triangle.

According to the diagram above and as per the Sine rule, the equation should go something like:

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

Solving Trigonometric Problems using the Sine Rule: The Steps

Step 1: Note down the values required for calculation from the given problem

Take a second peek at the Sine Rule equation provided above. Now answer this question:

How many known values do you need at the very least to find out an unknown value through the Sine Rule equation?

The answer is:

• One angle, and

• Two sides.

Remember, you CANNOT proceed with your solution without knowing the three things highlighted above (i.e., 1 angle and 2 sides). Both of these values should be provided in the question itself. Note them down in your answer sheet separately for an easy reference.

Note: It is heavily recommended to convert the given angles from degrees to radians for a flawless calculation. Here’s how you do it in a few simple steps.

Let the given angle is 60°.

According to the procedure of converting degrees to radians, you will have to do this:

π/x= given angle

or, π/x= 60°.

or, 180° /60° = x [since π = 180°]

or, x = 3

Therefore, the given angle in radian is π/x = π/3.

Step 2: Determine the value of the third unknown angle (if required)

To calculate the value of the 3rd unknown angle, you will have to know the value of the other two. The process is a simple subtraction of the sum of the two known angles from 180°.

z (unknown angle) = 180° + x + y (x and y are the two known angles)

Note: The 3rd angle doesn’t always have to be evaluated. In case it is directly opposite to the side that needs to be found through the Sine rule procedure, you will have to find out the value of it through the above-mentioned procedure.

Step 3: Choose the perfect combination of sides and angles required for calculation

To use the Sine rule equation in your solution, you need to choose the right combination of sides and angles from the given problem.

If you think you need to calculate the value of the third unknown angle for your equation, do so and move on to the next step.

Step 4: Substitute the values in their rightful places and solve the equation

Put the right values in the right places and solve the equation through the use of basic algebraic and trigonometric operations.

Unknown sine values such as sin 35°, sin 75° should be given in the problem itself. But there are a few standard values that you should know at the very least such as:

TRIGONOMETRIC ANGLES	VALUES
sin 0°	0
sin 30°	1/2
sin 45°	1/√2
sin 60°	√3/2
sin 90°	1

Example problem

Given figure:

You will have to evaluate the value of the side AC through the Sine rule technique.

Solution,

Given,

∠ABC= 27°

∠BAC= 123°

BC = 12 cm.

Since the angle opposite to AC is already known, there’s no need to find out the value of the third unknown angle. Let’s move on to our calculations now.

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

or, AC= (12 x 0.454)/ 0.838

or, AC= 6.5, which is the answer.

So you see the problems are not too difficult. With a bit of practice and determination, you can master this technique in no time. With that, we’ll sign off finally for the day. Hope you had a good and enlightening read.