Class 10 Mathematics – Trigonometry (Part
1)
Introduction to Trigonometry
What is Trigonometry?
Trigonometry is the branch of mathematics that studies the relationship between the angles
and sides of a right-angled triangle.
It is widely used in:
Engineering
Architecture
Navigation
Surveying
Physics
Astronomy
Right-Angled Triangle
A right-angled triangle has one angle equal to 90°.
For an angle θ (theta):
Hypotenuse – The longest side, opposite the 90° angle.
Opposite Side (Perpendicular) – The side opposite the angle θ.
Adjacent Side (Base) – The side next to the angle θ (excluding the hypotenuse).
Hypotenuse
/|
/ |
/ | Opposite
/θ |
/____|
Adjacent
Trigonometric Ratios
There are six basic trigonometric ratios.
, Ratio Formula
sin θ Opposite / Hypotenuse
cos θ Adjacent / Hypotenuse
tan θ Opposite / Adjacent
cosec θ Hypotenuse / Opposite
sec θ Hypotenuse / Adjacent
cot θ Adjacent / Opposite
Easy Memory Trick
SOH – CAH – TOA
S → Sin = Opposite / Hypotenuse
C → Cos = Adjacent / Hypotenuse
T → Tan = Opposite / Adjacent
Reciprocal Ratios
Ratio Reciprocal
sin θ cosec θ
cos θ sec θ
tan θ cot θ
Important Trigonometric Values
Angle sin θ cos θ tan θ
0° 0 1 0
30° 1/2 √3/2 1/√3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
90° 1 0 Not Defined
Reciprocal Values
Angle cosec θ sec θ cot θ
0° Not Defined 1 Not Defined
,Angle cosec θ sec θ cot θ
30° 2 2/√3 √3
45° √2 √2 1
60° 2/√3 2 1/√3
90° 1 Not Defined 0
Example 1
In a right triangle,
Opposite = 6 cm
Adjacent = 8 cm
Hypotenuse = 10 cm
Find all six trigonometric ratios.
Solution
sin θ = 6/10 = 3/5
cos θ = 8/10 = 4/5
tan θ = 6/8 = 3/4
cosec θ = 10/6 = 5/3
sec θ = 10/8 = 5/4
cot θ = 8/6 = 4/3
Example 2
If
sin θ = 3/5
Find cos θ and tan θ.
Solution
Using the Pythagorean theorem:
Opposite = 3
, Hypotenuse = 5
Adjacent = √(5² − 3²)
= √16
=4
Therefore,
cos θ = 4/5
tan θ = 3/4
Applications of Trigonometry
Trigonometry is used to:
Measure the height of buildings.
Find the distance between two places.
Calculate angles in construction.
Solve navigation problems.
Study waves and sound.
Design bridges and roads.
Important Formulas
Trigonometric Ratios
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Exam Tips
Learn the standard values (0°, 30°, 45°, 60°, 90°) by heart.
Use SOH–CAH–TOA to remember the formulas.
Draw a right triangle before solving problems.
1)
Introduction to Trigonometry
What is Trigonometry?
Trigonometry is the branch of mathematics that studies the relationship between the angles
and sides of a right-angled triangle.
It is widely used in:
Engineering
Architecture
Navigation
Surveying
Physics
Astronomy
Right-Angled Triangle
A right-angled triangle has one angle equal to 90°.
For an angle θ (theta):
Hypotenuse – The longest side, opposite the 90° angle.
Opposite Side (Perpendicular) – The side opposite the angle θ.
Adjacent Side (Base) – The side next to the angle θ (excluding the hypotenuse).
Hypotenuse
/|
/ |
/ | Opposite
/θ |
/____|
Adjacent
Trigonometric Ratios
There are six basic trigonometric ratios.
, Ratio Formula
sin θ Opposite / Hypotenuse
cos θ Adjacent / Hypotenuse
tan θ Opposite / Adjacent
cosec θ Hypotenuse / Opposite
sec θ Hypotenuse / Adjacent
cot θ Adjacent / Opposite
Easy Memory Trick
SOH – CAH – TOA
S → Sin = Opposite / Hypotenuse
C → Cos = Adjacent / Hypotenuse
T → Tan = Opposite / Adjacent
Reciprocal Ratios
Ratio Reciprocal
sin θ cosec θ
cos θ sec θ
tan θ cot θ
Important Trigonometric Values
Angle sin θ cos θ tan θ
0° 0 1 0
30° 1/2 √3/2 1/√3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
90° 1 0 Not Defined
Reciprocal Values
Angle cosec θ sec θ cot θ
0° Not Defined 1 Not Defined
,Angle cosec θ sec θ cot θ
30° 2 2/√3 √3
45° √2 √2 1
60° 2/√3 2 1/√3
90° 1 Not Defined 0
Example 1
In a right triangle,
Opposite = 6 cm
Adjacent = 8 cm
Hypotenuse = 10 cm
Find all six trigonometric ratios.
Solution
sin θ = 6/10 = 3/5
cos θ = 8/10 = 4/5
tan θ = 6/8 = 3/4
cosec θ = 10/6 = 5/3
sec θ = 10/8 = 5/4
cot θ = 8/6 = 4/3
Example 2
If
sin θ = 3/5
Find cos θ and tan θ.
Solution
Using the Pythagorean theorem:
Opposite = 3
, Hypotenuse = 5
Adjacent = √(5² − 3²)
= √16
=4
Therefore,
cos θ = 4/5
tan θ = 3/4
Applications of Trigonometry
Trigonometry is used to:
Measure the height of buildings.
Find the distance between two places.
Calculate angles in construction.
Solve navigation problems.
Study waves and sound.
Design bridges and roads.
Important Formulas
Trigonometric Ratios
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
Exam Tips
Learn the standard values (0°, 30°, 45°, 60°, 90°) by heart.
Use SOH–CAH–TOA to remember the formulas.
Draw a right triangle before solving problems.
