Step-by-Step Methods to Solve Triangles in Trigonometry
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).
FAQs on Trigonometry: How to Solve Triangles Easily
1. What does it mean to 'solve a triangle' in trigonometry?
To 'solve a triangle' means to find the measurements of all three of its sides and all three of its angles. To do this, you typically start with a few known pieces of information (like two sides and one angle, or two angles and one side) and use trigonometric principles to find the unknown values.
2. What is the first step to easily solve any triangle problem?
The first and most important step is to identify what type of triangle you are working with and what information is given. Determine if it is a right-angled triangle or an oblique (non-right-angled) triangle. Then, list the known sides and angles. This initial analysis dictates which formula or rule you should apply for the easiest solution.
3. What are the basic trigonometric ratios for solving a right-angled triangle?
For right-angled triangles, we use the three primary trigonometric ratios, often remembered by the mnemonic SOH CAH TOA. These ratios relate an angle (θ) to the lengths of the sides:
- Sine (sin θ) = Length of the Opposite Side / Length of the Hypotenuse
- Cosine (cos θ) = Length of the Adjacent Side / Length of the Hypotenuse
- Tangent (tan θ) = Length of the Opposite Side / Length of the Adjacent Side
4. When should you use the Sine Rule versus the Cosine Rule for non-right-angled triangles?
The choice depends on the information you have.
- Use the Sine Rule (a/sin A = b/sin B) when you know a side and its opposite angle (a 'pair'), along with one other side or angle. It is ideal for cases involving Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA).
- Use the Cosine Rule (c² = a² + b² - 2ab cos C) when you know two sides and the angle between them (Side-Angle-Side or SAS), or when you know all three sides (Side-Side-Side or SSS) and need to find an angle.
5. Why can't the basic SOH CAH TOA ratios be used to solve oblique (non-right-angled) triangles?
The SOH CAH TOA ratios are fundamentally defined by the properties of a right-angled triangle. The terms 'Hypotenuse' (the side opposite the right angle), 'Opposite', and 'Adjacent' only have clear, unambiguous meanings in the context of a 90° angle. In an oblique triangle, there is no hypotenuse, so these basic ratios do not apply. This is why the more general Sine and Cosine rules are required.
6. What is the '3-4-5 rule' in trigonometry and how is it a shortcut?
The '3-4-5 rule' refers to the most common example of a Pythagorean Triple. It states that if a triangle has sides in the ratio 3:4:5, it is guaranteed to be a right-angled triangle. This is a useful shortcut because if you recognise these side lengths (or their multiples, like 6-8-10 or 9-12-15), you immediately know it's a right triangle without needing to use the Pythagorean theorem to check, saving time in calculations.
7. What is the importance of the 'angle sum property' when solving triangles?
The angle sum property states that the sum of the three interior angles in any triangle is always 180 degrees. This is a crucial and simple tool for solving triangles. If you know any two angles in a triangle, you can instantly find the third angle by subtracting the sum of the known angles from 180°. This often provides the missing piece of information needed to then apply the Sine or Cosine rule.
8. What are some real-world examples where solving triangles is applied?
Solving triangles is a fundamental concept used in many real-world fields. For example:
- In architecture and engineering, it is used to calculate the lengths of beams, the height of structures, and the required angles for stable construction.
- In navigation and astronomy, it helps in pinpointing a location (triangulation), plotting a course for ships or aircraft, and measuring the vast distances between celestial bodies.
- In video game development and computer graphics, it is used to determine object positions, create realistic movements, and render 3D environments.
