Trigonometry - Solving Triangles

The trig ratios can be used to discover ample information, and one of their crucial purposes is to help solve triangles. This is used to solve a triangle that implies to calculate the length of all the sides as well as the measure of all the angles. It consists of a number of utilitarian tools such as the sin function and its inverse the arcsin function. This lesson will cover how to find the angle of a triangle and how to use trig ratios to calculate the side lengths of a triangle.

Solving Problems Using Trigonometry

Imagine trigonometry as a toolbox. Your task is to account for the problem and look at which tools can be used to obtain the answer.

We use a 5 step process for solving triangles, finding triangle sides and to solve other trigonometry problems:

• If a diagram is not provided, draw one yourself.

• Trace the right triangles.

• Pick the tool ‘solve the triangle calculator’ that leads to the answer.

• Use algebra for solving the problem.

• Check the answer if it looks reasonable.

Trigonometry Calculator to Solve Right Triangle

Trigonometry Calculator as a tool for solving triangles, to find an angle of a triangle or finding right triangle sides, all we require to do is to enter the known variables into the trigonometry calculator. You require only two given values in an instance of:

• One angle and one side.

• Two sides

• One side and area.

Remember that if you are aware of two angles, it's not adequate to determine the sides of the triangle. Two triangles consisting of the same shape (which states they have equal angles) may be of varying sizes (not the similar side length) - that kind of link is known as the triangle similarity. If the sides have a similar length, then the triangles are congruent.

Solved Examples For Finding Triangle Sides

Example 1:

Find b in the image given below.

Solution:

We will solve the following example step-by-step as below:

Step 1: Select which trig ratio to use.

Firstly, we know we should look at angle B since that is the angle we already know the

Measure of. (Now, you could find out the angle A and then use that but that would be less reliable because you could end up making an error.)

So, taking into account angle B, we want to determine which sides are involved. We are familiar that one side length is 8m, and that side is adjacent to angle B. The side of the triangle we're looking for is opposite to angle B. So we are required to select the trig ratio which is opposite and adjacent. This obviously is the tangent.

Step 2: Substitute

Next, we will write down our trig ratio:

Then, we substitute the side and the angle we know:

Step 3: Solve

Now moving the 8 to the other side by multiplying both sides by 8 we obtain: 3.7 m

Round to the nearest tenth, we get ‘b’ = 3.7 m.

Example 2:

Find c in the image given below.

Solution: 

You already know this is an extension to the above example.

Now that we are known with two sides, we could apply the Pythagorean Theorem in order to find the 3rd.  But again that would be less reliable considering that if we make a mistake on side b, then side c will also be incorrect.

Thus, we are going to repeat a similar process to find the length for side c.

Step 1: Select the trig ratio to use.

We're still taking into consideration the angle B.  The c is the hypotenuse and 8m is the adjacent.

The trig ratio that uses the hypotenuse and adjacent is the cosine.

Step 2: Substitute. Now mathematically write our trig ratio: 

Then, we substitute the side and the angle we know:

Step 3: Solve

because our variable is on the bottom, we can begin by cross multiplication: 

Then divide both the sides by cos 25°: 

Rounding to the nearest tenth, we get 8.8 m.

Note: If your calculator doesn't seem to be giving you the correct answer, use the Vedantu assisted ‘solve the triangle calculator’ for solving triangles. Just make sure your calculator is in the mode of Degree (not Radians).