TRIGONOMETRY
Complete Concept • Definitions • Essential Formulas • Quick Revision
1. What is Trigonometry?
Trigonometry is the branch of mathematics that studies the relationship between angles and sides of triangles. It is used in geometry,
navigation, architecture, physics, surveying, waves, and many real-life measurements.
2. Basic Terms
Angle: A rotation between two rays. Common units are degrees (°) and radians (rad). Right angle: 90° = π/2 rad.
Hypotenuse (H): The longest side of a right triangle, opposite the 90° angle. Opposite (O): The side across from the chosen angle
θ. Adjacent (A): The side next to θ, other than the hypotenuse.
3. The Six Trigonometric Ratios (SOH–CAH–TOA)
Ratio Definition Memory clue
sin θ O/H SOH: Sine = Opposite / Hypotenuse
cos θ A/H CAH: Cosine = Adjacent / Hypotenuse
tan θ O/A TOA: Tangent = Opposite / Adjacent
cosec θ H / O = 1/sin θ Reciprocal of sine
sec θ H / A = 1/cos θ Reciprocal of cosine
cot θ A / O = 1/tan θ Reciprocal of tangent
Important: For a fixed acute angle, sin, cos and tan are ratios of side lengths. The chosen angle determines which side is opposite
and which is adjacent.
TRIGONOMETRY • COMPLETE SHORT NOTES Page 1 of 5
, PAGE 2 • CORE IDENTITIES & RELATIONSHIPS
4. Reciprocal and Quotient Identities
cosec θ = 1/sin θ | sec θ = 1/cos θ | cot θ = 1/tan θ
tan θ = sin θ / cos θ | cot θ = cos θ / sin θ
sin θ × cosec θ = 1 | cos θ × sec θ = 1 | tan θ × cot θ = 1
5. Pythagorean Identities
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
Rearrangements: sin² θ = 1 − cos² θ; cos² θ = 1 − sin² θ; sec² θ − tan² θ = 1; cosec² θ − cot² θ = 1.
6. Complementary-Angle Formulas
sin(90° − θ) = cos θ | cos(90° − θ) = sin θ
tan(90° − θ) = cot θ | cot(90° − θ) = tan θ
sec(90° − θ) = cosec θ | cosec(90° − θ) = sec θ
7. Standard Angle Values
θ 0° 30° 45° 60° 90°
sin θ 0 1/2 1/√2 √3/2 1
cos θ 1 √3/2 1/√2 1/2 0
tan θ 0 1/√3 1 √3 Undefined
For cosec, sec and cot, take reciprocals of sin, cos and tan respectively; values may be undefined when the denominator is zero.
TRIGONOMETRY • COMPLETE SHORT NOTES Page 2 of 5
Complete Concept • Definitions • Essential Formulas • Quick Revision
1. What is Trigonometry?
Trigonometry is the branch of mathematics that studies the relationship between angles and sides of triangles. It is used in geometry,
navigation, architecture, physics, surveying, waves, and many real-life measurements.
2. Basic Terms
Angle: A rotation between two rays. Common units are degrees (°) and radians (rad). Right angle: 90° = π/2 rad.
Hypotenuse (H): The longest side of a right triangle, opposite the 90° angle. Opposite (O): The side across from the chosen angle
θ. Adjacent (A): The side next to θ, other than the hypotenuse.
3. The Six Trigonometric Ratios (SOH–CAH–TOA)
Ratio Definition Memory clue
sin θ O/H SOH: Sine = Opposite / Hypotenuse
cos θ A/H CAH: Cosine = Adjacent / Hypotenuse
tan θ O/A TOA: Tangent = Opposite / Adjacent
cosec θ H / O = 1/sin θ Reciprocal of sine
sec θ H / A = 1/cos θ Reciprocal of cosine
cot θ A / O = 1/tan θ Reciprocal of tangent
Important: For a fixed acute angle, sin, cos and tan are ratios of side lengths. The chosen angle determines which side is opposite
and which is adjacent.
TRIGONOMETRY • COMPLETE SHORT NOTES Page 1 of 5
, PAGE 2 • CORE IDENTITIES & RELATIONSHIPS
4. Reciprocal and Quotient Identities
cosec θ = 1/sin θ | sec θ = 1/cos θ | cot θ = 1/tan θ
tan θ = sin θ / cos θ | cot θ = cos θ / sin θ
sin θ × cosec θ = 1 | cos θ × sec θ = 1 | tan θ × cot θ = 1
5. Pythagorean Identities
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
Rearrangements: sin² θ = 1 − cos² θ; cos² θ = 1 − sin² θ; sec² θ − tan² θ = 1; cosec² θ − cot² θ = 1.
6. Complementary-Angle Formulas
sin(90° − θ) = cos θ | cos(90° − θ) = sin θ
tan(90° − θ) = cot θ | cot(90° − θ) = tan θ
sec(90° − θ) = cosec θ | cosec(90° − θ) = sec θ
7. Standard Angle Values
θ 0° 30° 45° 60° 90°
sin θ 0 1/2 1/√2 √3/2 1
cos θ 1 √3/2 1/√2 1/2 0
tan θ 0 1/√3 1 √3 Undefined
For cosec, sec and cot, take reciprocals of sin, cos and tan respectively; values may be undefined when the denominator is zero.
TRIGONOMETRY • COMPLETE SHORT NOTES Page 2 of 5
