MATHEMATICS • TRIGONOMETRY • CLASSES 10–12
What Is Trigonometry? Complete Notes, Ratios, Identities and Real-Life Uses
Trigonometry connects the angles of a triangle with the lengths of its sides. It helps us calculate heights, distances, slopes, directions and repeating patterns that are difficult to measure directly.
This guide explains the six ratios, standard-angle table, identities, real-world applications and solved exam-style questions.
1. What Is Trigonometry?
Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles. In elementary geometry, it mainly uses ratios in right-angled triangles. At higher levels, it also studies rotations, waves and periodic phenomena.
For a fixed acute angle, the ratios of corresponding sides remain constant in similar triangles. Therefore, if an angle and one side are known, a suitable trigonometric ratio can help calculate an unknown side.
Uses in real life
- Finding the height of a building, tree, tower or mountain.
- Calculating distances across a river or between inaccessible points.
- Surveying, navigation, mapping and astronomy.
- Designing roofs, ramps, bridges and other structures.
- Physics: forces, sound, light, waves and circular motion.
- Computer graphics, animation, robotics, games and satellite positioning.
2. Understand the Right-Angled Triangle
- Hypotenuse (H): side opposite the right angle; always the longest side.
- Opposite (O): side directly opposite the chosen reference angle.
- Adjacent (A): side next to the chosen angle, other than the hypotenuse.
3. All Six Trigonometric Ratios
For an acute angle θ in a right-angled triangle:
sin θ = Opposite / Hypotenuse = O/H
cos θ = Adjacent / Hypotenuse = A/H
tan θ = Opposite / Adjacent = O/A
cosec θ = Hypotenuse / Opposite = H/O
sec θ = Hypotenuse / Adjacent = H/A
cot θ = Adjacent / Opposite = A/O
| Ratio | Meaning | Formula | Reciprocal relation |
|---|---|---|---|
| sin θ | Sine | O/H | cosec θ = 1/sin θ |
| cos θ | Cosine | A/H | sec θ = 1/cos θ |
| tan θ | Tangent | O/A | cot θ = 1/tan θ |
| cosec θ | Cosecant | H/O | sin θ = 1/cosec θ |
| sec θ | Secant | H/A | cos θ = 1/sec θ |
| cot θ | Cotangent | A/O | tan θ = 1/cot θ |
Remember: SOH–CAH–TOA
SOH: Sine = Opposite/Hypotenuse.
CAH: Cosine = Adjacent/Hypotenuse.
TOA: Tangent = Opposite/Adjacent.
Example: a 3–4–5 triangle
If opposite = 3, adjacent = 4 and hypotenuse = 5, then sin θ = 3/5, cos θ = 4/5, tan θ = 3/4, cosec θ = 5/3, sec θ = 5/4 and cot θ = 4/3.
4. Trigonometric Table of Standard Angles
Memorise these exact values for 0°, 30°, 45°, 60° and 90°.
| 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 | Not defined |
| cosec θ | Not defined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | Not defined |
| cot θ | Not defined | √3 | 1 | 1/√3 | 0 |
Pattern for sine and cosine
For sine, use √0/2, √1/2, √2/2, √3/2, √4/2 in order.
Thus sin θ = 0, 1/2, 1/√2, √3/2, 1. The cosine row is the same list in reverse order.
A ratio with denominator zero is undefined. For example, tan 90° = sin 90°/cos 90° = 1/0 is undefined, whereas cot 90° = 0.
5. Ratios of Complementary Angles
Complementary angles add up to 90°. If A + B = 90°, then:
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
For example, sin 30° = cos 60° = 1/2, and tan 30° = cot 60° = 1/√3.
6. Signs in the Four Quadrants
For angles measured around coordinate axes, the signs depend on the quadrant. “All Students Take Calculus” is a common memory aid.
| Quadrant | Positive ratios | Memory aid |
|---|---|---|
| I | All six | All |
| II | sin, cosec | Students (sine) |
| III | tan, cot | Take (tangent) |
| IV | cos, sec | Calculus (cosine) |
7. Trigonometric Identities
A trigonometric identity is an equality involving trigonometric functions that is true for every angle in the common domain where both sides are defined. Identities are used to simplify expressions, calculate unknown ratios and prove statements.
7.1 Reciprocal identities
cosec θ = 1/sin θ; sin θ = 1/cosec θ
sec θ = 1/cos θ; cos θ = 1/sec θ
cot θ = 1/tan θ; tan θ = 1/cot θ
7.2 Quotient identities
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
7.3 Fundamental Pythagorean identities
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
In sin² θ, the exponent applies to the value of sine: sin² θ means (sin θ)², not sin(θ²).
7.4 Rearranged forms
sin² θ = 1 − cos² θ
cos² θ = 1 − sin² θ
tan² θ = sec² θ − 1
sec² θ − tan² θ = 1
cot² θ = cosec² θ − 1
cosec² θ − cot² θ = 1
7.5 Complementary-angle identities
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
7.6 Sum and difference formulae (higher-level)
sin (A + B) = sin A cos B + cos A sin B
sin (A − B) = sin A cos B − cos A sin B
cos (A + B) = cos A cos B − sin A sin B
cos (A − B) = cos A cos B + sin A sin B
tan (A + B) = (tan A + tan B)/(1 − tan A tan B)
tan (A − B) = (tan A − tan B)/(1 + tan A tan B)
The tangent formulae require all terms to be defined and the displayed denominators to be non-zero.
7.7 Double-angle formulae
sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A
cos 2A = 1 − 2 sin² A = 2 cos² A − 1
tan 2A = 2 tan A / (1 − tan² A)
7.8 Half-angle formulae
sin² (A/2) = (1 − cos A)/2
cos² (A/2) = (1 + cos A)/2
tan² (A/2) = (1 − cos A)/(1 + cos A)
When taking square roots, choose the positive or negative sign according to the quadrant of A/2.
7.9 Product-to-sum and sum-to-product formulae
2 sin A cos B = sin (A + B) + sin (A − B)
2 cos A sin B = sin (A + B) − sin (A − B)
2 cos A cos B = cos (A + B) + cos (A − B)
2 sin A sin B = cos (A − B) − cos (A + B)
sin A + sin B = 2 sin ((A + B)/2) cos ((A − B)/2)
sin A − sin B = 2 cos ((A + B)/2) sin ((A − B)/2)
cos A + cos B = 2 cos ((A + B)/2) cos ((A − B)/2)
cos A − cos B = −2 sin ((A + B)/2) sin ((A − B)/2)
8. Degrees and Radians
Angles may be measured in degrees or radians. A full revolution is 360° or 2π radians, so 180° = π radians.
Degrees to radians: θ° × π/180
Radians to degrees: θ × 180/π
| Degrees | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
Example: convert 150° into radians
150 × π/180 = 5π/6 radians.
Example: convert 2π/3 radians into degrees
(2π/3) × (180/π) = 120°.
8.1 The unit-circle idea
On a unit circle of radius 1 centred at the origin, the point corresponding to angle θ has coordinates (cos θ, sin θ). This explains why cosine is the horizontal coordinate, sine is the vertical coordinate, and sin² θ + cos² θ = 1.
9. Uses of Trigonometry in Real Life
9.1 Finding the height of a building or tree
If horizontal distance is d and angle of elevation is θ, ignoring observer height:
tan θ = height/d ⇒ height = d tan θ
9.2 Elevation and depression
- Angle of elevation: angle between the horizontal and line of sight when looking upward.
- Angle of depression: angle between the horizontal and line of sight when looking downward.
When the horizontal lines are parallel, the angle of depression at the top and the angle of elevation at the bottom are equal as alternate interior angles.
9.3 Other practical uses
Surveying: estimate a tower's height or the width of a river using measured baselines and angles.
Construction: calculate roof slopes, ramp gradients, stairs and bridge components. Engineering designs also require safety factors and material checks.
Navigation and astronomy: describe bearings, satellite geometry and positions of celestial objects.
Physics and technology: analyse forces, sound waves, rotating machinery, animation, robotics and movement in computer graphics.
10. Solved Examples
Opposite = 5 cm and hypotenuse = 13 cm. Find sin θ and cos θ.
Adjacent² = 13² − 5² = 144, so adjacent = 12 cm. Therefore sin θ = 5/13 and cos θ = 12/13.
Evaluate 2 sin 30° + cos 60°.
= 2(1/2) + 1/2 = 1 + 1/2 = 3/2.
Simplify (1 − cos² θ)/sin² θ.
Since 1 − cos² θ = sin² θ, the expression equals 1, wherever defined.
A student stands 20 m from a pole. The angle of elevation is 45°. Find the pole's height, ignoring eye height.
tan 45° = h/20. Since tan 45° = 1, h = 20 m.
From 10 m away, the angle of elevation to a tower is 60°. Find its height.
tan 60° = h/10, so h = 10√3 m ≈ 17.32 m.
If sin θ = 3/5 and θ is acute, find cos θ and tan θ.
Opposite = 3, hypotenuse = 5, so adjacent = 4. Hence cos θ = 4/5 and tan θ = 3/4.
Prove sec² θ − tan² θ = 1.
Divide sin² θ + cos² θ = 1 by cos² θ: tan² θ + 1 = sec² θ. Rearranging gives the result.
11. Common Mistakes to Avoid
- Confusing opposite and adjacent: identify them relative to the marked angle.
- Forgetting that the hypotenuse is always opposite the right angle.
- Mixing up sine and cosine values. The cosine row reverses the sine row for the standard angles.
- Writing tan 90° = 0. It is undefined; cot 90° = 0.
- Confusing reciprocal ratios with negative values: cosec θ = 1/sin θ.
- Dividing an identity by an expression that may be zero without checking its domain.
- Forgetting observer's eye height if a height-and-distance question includes it.
12. Exam Practice Questions
Try each question before revealing the answer. These are original practice questions based on common school-level concepts, not verbatim copies of a specific board paper.
Q1. Write tan θ in terms of sin θ and cos θ.
Show answer
tan θ = sin θ/cos θ, where cos θ ≠ 0.
Q2. Find cos 60°.
Show answer
1/2.
Q3. If tan θ = 3/4 for an acute angle, find sin θ.
Show answer
Opposite = 3, adjacent = 4, hypotenuse = 5. Therefore sin θ = 3/5.
Q4. Evaluate sin² 30° + cos² 30°.
Show answer
By sin² θ + cos² θ = 1, the answer is 1.
Q5. Simplify (sec² θ − 1)/tan² θ.
Show answer
Since sec² θ − 1 = tan² θ, the value is 1 wherever defined.
Q6. Prove (1 − sin² θ)/cos² θ = 1.
Show answer
Using 1 − sin² θ = cos² θ, the left side becomes cos² θ/cos² θ = 1, wherever defined.
Q7. A 10 m ladder makes an angle of 60° with the ground. Find the height reached.
Show answer
Height = 10 sin 60° = 10√3/2 = 5√3 m ≈ 8.66 m.
Q8. Convert 225° into radians.
Show answer
225 × π/180 = 5π/4 radians.
Q9. A tower casts a 12 m shadow when the sun's angle of elevation is 30°. Find its height.
Show answer
tan 30° = h/12. Thus h = 12/√3 = 4√3 m ≈ 6.93 m.
Q10. If sec θ = 13/12 and θ is acute, find tan θ.
Show answer
tan² θ = sec² θ − 1 = 169/144 − 1 = 25/144. Since θ is acute, tan θ = 5/12.
Q11. Prove (sec θ − 1)(sec θ + 1) = tan² θ.
Show answer
Left side = sec² θ − 1 = tan² θ, using the fundamental identity.
13. Quick Revision Sheet
Six ratios: sin = O/H, cos = A/H, tan = O/A, cosec = H/O, sec = H/A, cot = A/O.
Core identities: sin² θ + cos² θ = 1; 1 + tan² θ = sec² θ; 1 + cot² θ = cosec² θ.
Reciprocal pairs: sin–cosec, cos–sec, tan–cot.
Quotients: tan θ = sin θ/cos θ; cot θ = cos θ/sin θ.
Complementary angles: sin(90° − θ) = cos θ; tan(90° − θ) = cot θ.
Degrees/radians: 180° = π radians.
Height and distance: tan θ = height/horizontal distance for the illustrated right triangle.
For more mathematics notes, visit the Saraswat Academy Knowledge Vault and select Mathematics.
Reference Resources
- NCERT textbook portal — choose Mathematics for your class.
- NCERT Class 10 Mathematics textbook contents — includes Introduction to Trigonometry and Some Applications of Trigonometry.
- CBSE Curriculum 2026–27 — verify the syllabus for your class and subject.
Final advice: draw a labelled triangle, choose the correct ratio, substitute the known values, show each calculation and include units in real-life problems.