Trigonometry Solved Examples
Example 1: A ladder is leaning against a wall. The angle between the ladder and the ground is 45 degrees, and the length of the ladder is 10 meters. How far is the ladder from the wall?
Solution:
Let the distance between the ladder and the wall be x meters.
Here, ladder, wall and ground together makes a right angle triangle, where for given angle,
Length of ladder = hypotenous = 10 meter,
Distance between wall and laddar = base = x meterUsing trigonometric ratio cos, we get
⇒ cos(45°) = = x/10
⇒ cos(45°) = 1
⇒1/√2 = x/10
⇒ x = 10/√2 = 5√2 meters
Therefore, the ladder is 5√2 meters away from the wall.
Example 2: A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles measures 30°. What are the lengths of the other two sides?
Solution:
Let’s call the side opposite to the 30° angle as ‘a’ and the side adjacent to it as ‘b’.
Now, sin (30°) = perpendicular/hypotenous = a/10
⇒ a = 10 × sin(30°) = 5 cm [sin(30°) = 1/2]
and cos(30°) = b/10
⇒ b = 10 × cos(30°) = 10 × √(3)/2 ≈ 8.66 cm
Therefore, the lengths of the other two sides are 5 cm and 8.66 cm (approx.).
Example 3: Prove that (cos x/sin x) + (sin x/cos x) = sec x × cosec x.
Solution:
LHS = (cos x/sin x) + (sin x/cos x)
⇒ LHS = [cos2x + sin2x]/(cos x sin x)
⇒ LHS = 1/(cosx sinx) [Using cos2x + sin2x = 1]
⇒ LHS = (1/cosx) × (1/sinx)
⇒ LHS = secx × cosecx = RHS [ 1/cosx = sec x and 1/sinx = cosec x]
Example 4: A person is standing at a distance of 10 meters from the base of a building. The person measures the angle of elevation to the top of the building as 60°. What is the height of the building?
Solution:
Let h be the height of the building.
and, all the distances here in the question make a right angle triangle, with a base of 10 meters and height h meter.
As tan θ = Perpendicular/Base
⇒ tan(60°) = h / 10
⇒ h = 10 tan(60°)
Using the values of tan(60°) = √3, we get:
h = 10√3 ≈ 17.32 m
Therefore, the height of the building is approximately 17.32 meters.
Example 5: Find the value of x in the equation cos-1(x) + sin-1(x) = π/4.
Solution:
For, cos-1(x) + sin-1(x) = π/4
As we know, cos-1(x) + sin-1(x) = π/2, above equation becomes
π/2 = π/4, which is not true.
Thus, the given equation has no such value of x, which can satisfy the equation.
Trigonometry in Maths: Table, Formulas, Identities and Ratios
Trigonometry is a branch of mathematics that explores the relationships between the ratios of the sides of a right-angled triangle and its angles. The fundamental ratios used to study these relationships are known as trigonometric ratios, which include sine, cosine, tangent, cotangent, secant, and cosecant.
The term “trigonometry” is a 16th-century Latin derivative and the concept was given by the Greek mathematician Hipparchus. Trigonometry word is formed from ancient Greek words “trigonon” and “metron” which mean triangle and measure respectively, thus collectively called Trigonometry which means measures of a triangle.
The most important topics in trigonometry are trigonometry table, trigonometry formulas, trigonometric identities, and trigonometric ratios. In this article, we will see the basics of trigonometry, including its fundamental identities and formulas.
Table of Content
- Introduction to Trigonometry
- What is Trigonometry?
- Trigonometry Definition
- Trigonometry Basics
- All Trigonometry Functions
- Even and Odd Trigonometric Functions
- Odd Trigonometric Functions
- Even Trigonometric Functions
- Trigonometric Ratios
- Trigonometry Angles
- Trigonometry Chart
- Trigonometry Table
- How to Remember Trigonometry Table?
- Applications of Trigonometry
- Trigonometry Formulas
- 1. Pythagorean Trigonometric Identities
- 2. Sum and Difference Identities
- 3. Double angle Identities
- 4. Half Angle Identities
- 5. Product Sum Identities
- 6. Product Identities
- 7. Triple Angle Identities
- Unit Circle
- Trigonometry Identities
- Euler’s Formula for Trigonometry
- Trigonometry Real-Life Examples
- Trigonometry Solved Examples
- Practice Problems on Trigonometry
- Trigonometry Class 10 PDF