Solving for a Side in Right Triangles with Trigonometric Ratio
This is one of the most basic and useful uses of trigonometry using the trigonometric ratios mentioned to find the length of a side of a right-angled triangle but to do, so we must already know the length of the other two sides or an angle and length of one side.
Steps to follow if one side and one angle are known:
- Choose a trigonometric ratio that contains the given side and the unknown side
- Use algebra to find the unknown side
Example: In a right-angled triangle, ABC ∠B = 90° and ∠C = 30° length of side AB is 4 find the length of BC given tan 30° = 1/√3.
Solution:
C = 30°
tan C = tan 30°
= 1/√3
tan C = opposite side/adjacent side
1/√3 = AB/BC
1/√3 = 4/BC
BC = 4√3
Steps to follow if two sides are known:
- Mark the known sides as adjacent, opposite, or hypotenuse with respect to any one of the acute angles in the triangle.
- Decide on which trigonometric ratio can be found from the above table.
- Find the angle (X)
- Use a trigonometric ratio with respect to X which is a ratio of a known side and an unknown side.
- Use algebra to find the unknown side.
Example: If two sides of a right-angled triangle are 20 and 10√3 where the side with length 20 is the hypotenuse, find the third side (without using Pythagoras theorem) given sin 30° = 1/2 and cos 30° = √3/2
Solution:
Given:
Hypotenuse =20, one side = 10√3, sin30° = 1/2, cos30° = √3/2
we can deduce that the angle opposite the side of length 10√3 is 30°, as sin 30° = 1/2 and cos30° = √3/2 are the values associated with 30°
using sine ratio:
sin(30°) = opposite/Hypotenuse
1/2 = opposite/20
Solving for opposite side, we get:
opposite = (1/2) × 20 = 10
So, length of third side of the triangle is 10 units.
Articles related to Trigonometric Ratios:
Trigonometric Ratios
Trigonometric Ratios: There are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratio between these sides based on the angle between them is called Trigonometric Ratio.
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). Trigonometry, a branch of mathematics, focuses on the relationships between the sides and angles of right-angled triangles. Consequently, trigonometric ratios are determined based on these sides and angles.
Table of Content
- What are Trigonometric Ratios?
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Cosecant (cosec)
- Secant (sec)
- Cotangent (cot)
- Trigonometric Ratios Table
- Trigonometric Table of Some Specific Angles
- Solving for a Side in Right Triangles with Trigonometric Ratio
- Trigonometric Ratios Examples
- Practice Problems on Trigonometric Ratios
As given in the figure in a right-angle triangle
- The side opposite the right angle is called the hypotenuse
- The side opposite to an angle is called the opposite side
- For angle C opposite side is AB
- For angle A opposite side is BC
- The side adjacent to an angle is called the adjacent side
- For angle C adjacent side is BC
- For angle A adjacent side is AB