Grade 10 Notes of Trigonometry || Compulsory Mathematics

1.Right-Angled Triangle

Definition:
A triangle with one angle equal to 90° is called a right-angled triangle.

Let triangle ABC be right-angled at C.

Key Elements:
1.1 Hypotenuse = Side opposite the right angle (longest side)
1.2 Perpendicular = Side opposite angle θ
1.3 Base = Side adjacent to angle θ

Formulas:
1.4 Area of Right-Angled Triangle: Area = 1/2 × base × height
1.5 Pythagoras Theorem: Hypotenuse^2 = Base^2 + Perpendicular^2
1.6 Trigonometric Ratios (for angle θ):
sin θ = Perpendicular / Hypotenuse
cos θ = Base / Hypotenuse
tan θ = Perpendicular / Base

Common Values of Trigonometric Ratios:

Example:
AB = 6 cm, AC = 10 cm, find BC using Pythagoras theorem:
BC^2 + AB^2 = AC^2
BC^2 + 6^2 = 10^2 ⇒ BC = 8 cm

Trigonometric ratios for ∠A:
sin A = BC / AC = 8/10 = 0.8
cos A = AB / AC = 6/10 = 0.6
tan A = BC / AB = 8/6 = 4/3

2.Angle of Elevation and Depression

Definitions:
2.1 Angle of Elevation = Angle formed by line of sight above horizontal when looking up
2.2 Angle of Depression = Angle formed by line of sight below horizontal when looking down

Key Points:
2.3 Draw a horizontal line from observer’s eye
2.4 Angle of elevation = angle between line of sight and horizontal (looking up)
2.5 Angle of depression = angle between line of sight and horizontal (looking down)
2.6 Measured using clinometer

Observation:
2.7 Closer object → larger angle of elevation
2.8 Farther object → smaller angle of elevation

3.Applications of Trigonometry

3.1 Right-Triangle Relationships:
tan θ = (Height of object − Observer height) / Distance from object
sin θ = (Height of object − Observer height) / Hypotenuse

Key Formulas:
3.2 sin θ = Opposite / Hypotenuse
3.3 cos θ = Adjacent / Hypotenuse
3.4 tan θ = Opposite / Adjacent
3.5 Height of object = (tan θ × distance) + observer height
3.6 Distance of object = (Height of object − observer height) / tan θ
3.7 Pythagoras theorem: Hypotenuse^2 = Perpendicular^2 + Base^2
3.8 Angle from shadow: tan θ = Height of object / Shadow length

4.Solved Examples (Questions)

4.1 What is the height of a tree if the angle of elevation is 45°, the distance from the tree is 20 m, and the observer’s height is 1.8 m?
tan 45° = (h − 1.8)/20 ⇒ h = 21.8 m

4.2 What is the distance to a tower if its height is 140 m and the angle of elevation is 60°?
tan 60° = 140 / x ⇒ x = 80.83 m

4.3 If a tree is broken and the top touches the ground, what is the distance from the base if the height is 18 m and the angle is 30°?
sin 30° = (18 − x)/x ⇒ x = 12 m

4.4 What is the angle formed by a pole in a circular pond if the height is 50 m and the radius of the pond is 50 m?
tan θ = 50 / 50 ⇒ θ = 45°

4.5 What is the height of a house if a tower of height 60 m has an angle of depression of 45° from the top and the distance to the house is 20 m?
tan 45° = (60 − h)/20 ⇒ h = 40 m

4.6 What is the distance between two towers if their heights are 50 m and 30 m, with angles of elevation 45° and 60° from a point?
tan 45° = 50 / x ⇒ x = 50 m
tan 60° = 30 / y ⇒ y = 30/√3 ≈ 17.32 m

4.7 How high is a building if its shadow is 24 m and the angle of elevation of the top is 30°?
tan 30° = h / 24 ⇒ h = 24/√3 ≈ 13.86 m

4.8 An observer of height 2 m sees the top of a tower at angles 45° and 60° when walking 10 m closer. What is the height of the tower?
tan 45° = (h − 2)/(x + 10) ⇒ h − 2 = x + 10
tan 60° = (h − 2)/x ⇒ h − 2 = x√3
Solve: x ≈ 17.32 m, h ≈ 32 m

4.9 What is the width of a river if two points on opposite banks see a tree of 20 m height at angles of elevation 30° and 60°?
tan 30° = 20 / x ⇒ x ≈ 34.64 m
tan 60° = 20 / y ⇒ y ≈ 11.55 m
Width = x + y ≈ 46.19 m

4.10 How far is a building on a hill if the building is 50 m, hill height = 30 m, and angles of elevation are 60° and 45°?
tan 45° = 30 / x ⇒ x = 30 m
tan 60° = 80 / x ⇒ x ≈ 46.19 m

4.11 How far is a ship from a lighthouse of height 40 m if the angles of elevation are 30° and 45° after sailing 20 m closer?
tan 30° = 40 / (x + 20) ⇒ x ≈ 49.28 m

4.12 What is the angle of inclination if a pole of height 20 m leans towards another pole of 15 m at a distance of 10 m?
tan θ = 5 / 10 ⇒ θ ≈ 26.57°

4.13 How far is a plane if its height is 500 m and the angle of elevation from a point is 30°?
tan 30° = 500 / x ⇒ x ≈ 866.03 m

4.14 What is the distance between two boats observed from a cliff of height 60 m with angles of depression 30° and 45°?
Distance to first boat = 60 / tan 30° ≈ 103.92 m
Distance to second boat = 60 / tan 45° = 60 m
Distance between boats ≈ 43.92 m

4.15 If a tree of height 18 m breaks and its top touches the ground 12 m from the base, what is the broken height?
√(x^2 + 12^2) = 18 − x ⇒ x = 9 m

4.16 Two towers of heights 60 m and 40 m are observed. The observer is 50 m from the first tower, and the angle to the second tower is 30°. What is the distance between them?
tan 30° = 40 / (d − 50) ⇒ d ≈ 73.09 m

Practice Link:
https://besidedegree.com/exam/s/academic