All Important Trigonometry Formulas for Class 10
- Mar 16
- 8 min read
Trigonometry is one of the most important and scoring topics in Class 10 Mathematics (CBSE). Introduced formally in Chapter 8 and Chapter 9 of the NCERT textbook, trigonometry forms the foundation for higher-level mathematics, physics, and engineering. Mastering all the important trigonometry formulas for Class 10 can help students score full marks in board examinations and competitive entrance tests.
The word "Trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure). It is the branch of mathematics that deals with the relationships between the sides and angles of a right-angled triangle. In Class 10, students are introduced to six fundamental trigonometric ratios and a wide range of formulas derived from them.
This article covers all important trigonometry formulas for Class 10 in a structured, easy-to-understand format, including trigonometric ratios, standard angle values, reciprocal identities, Pythagorean identities, complementary angle formulas, and application-based formulas. Bookmark this page as your go-to revision resource before your board exams!
What is Trigonometry? (Class 10 Overview)
In Class 10 CBSE Trigonometry, the focus is entirely on right-angled triangles. For any right-angled triangle ABC, where angle C = 90°, the trigonometric ratios are defined with respect to one of the acute angles (say angle A).
The three sides of the right triangle are named as:
• Hypotenuse (H): The side opposite to the right angle, the longest side of the triangle.
• Perpendicular / Opposite (P): The side directly opposite to the angle being considered (angle A).
• Base / Adjacent (B): The side adjacent to the angle being considered (angle A), other than the hypotenuse.
The Six Trigonometric Ratios | Core Formulas
The six fundamental trigonometric ratios are the building blocks of all trigonometry formulas. Every student must memorize these before moving to advanced topics.
Trigonometric Ratio | Formula (in terms of Perpendicular, Base, Hypotenuse) |
sin A (Sine) | sin A = Perpendicular / Hypotenuse (P/H) |
cos A (Cosine) | cos A = Base / Hypotenuse (B/H) |
tan A (Tangent) | tan A = Perpendicular / Base (P/B) |
cosec A (Cosecant) | cosec A = Hypotenuse / Perpendicular (H/P) |
sec A (Secant) | sec A = Hypotenuse / Base (H/B) |
cot A (Cotangent) | cot A = Base / Perpendicular (B/P) |
Pro Tip: Use the mnemonic "Some People Have Curly Black Hair" to remember sin = P/H, cos = B/H, tan = P/B.
Reciprocal Identities | Must-Know Trigonometry Formulas
The reciprocal relationships between the six trigonometric ratios are among the most frequently tested formulas in Class 10 board exams.
Identity | Formula |
Cosecant | cosec A = 1 / sin A |
Secant | sec A = 1 / cos A |
Cotangent | cot A = 1 / tan A |
Sine | sin A = 1 / cosec A |
Cosine | cos A = 1 / sec A |
Tangent | tan A = 1 / cot A |
Important: Always remember that sin A × cosec A = 1, cos A × sec A = 1, and tan A × cot A = 1. These are quick verification tools during problem-solving.
Quotient Identities
These formulas express tan A and cot A in terms of sin and cos, and are extremely useful in simplifying expressions.
Identity | Formula |
Tangent in terms of sin and cos | tan A = sin A / cos A |
Cotangent in terms of sin and cos | cot A = cos A / sin A |
Pythagorean Identities | The Heart of Trigonometry
The Pythagorean identities are derived from the Pythagorean Theorem and are the most widely used formulas in solving Class 10 trigonometry problems. These appear in almost every board exam question.
Identity Number | Pythagorean Identity |
Identity 1 (Primary) | sin²A + cos²A = 1 |
Derived from Identity 1 (a) | sin²A = 1 - cos²A |
Derived from Identity 1 (b) | cos²A = 1 - sin²A |
Identity 2 | sec²A - tan²A = 1 |
Derived from Identity 2 (a) | sec²A = 1 + tan²A |
Derived from Identity 2 (b) | tan²A = sec²A - 1 |
Identity 3 | cosec²A - cot²A = 1 |
Derived from Identity 3 (a) | cosec²A = 1 + cot²A |
Derived from Identity 3 (b) | cot²A = cosec²A - 1 |
Exam Tip: In most proof-based questions and simplification problems, applying the Pythagorean identity sin²A + cos²A = 1 is the first step. Always keep an eye out for expressions that can be replaced using these identities.
Trigonometric Values of Standard Angles
One of the most critical sections to memorize is the trigonometric values of standard angles. These are angles for which exact values of all six trig ratios are known. Class 10 students must know values for 0°, 30°, 45°, 60°, and 90° by heart.
Angle | 0° | 30° | 45° | 60° | 90° |
sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
tan | 0 | 1/√3 | 1 | √3 | Undefined |
cosec | Undefined | 2 | √2 | 2/√3 | 1 |
sec | 1 | 2/√3 | √2 | 2 | Undefined |
cot | Undefined | √3 | 1 | 1/√3 | 0 |
Memory Trick for sin values (0° to 90°): sin 0° = √0/2, sin 30° = √1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = √4/2. The pattern is √0, √1, √2, √3, √4 divided by 2!
Memory Trick for cos values: The cosine row is simply the sine row in reverse order: 1, √3/2, 1/√2, 1/2, 0.
Complementary Angle Formulas — Trigonometry Class 10
Two angles are complementary if they add up to 90°. In a right triangle, the two acute angles are always complementary. The following complementary angle trigonometry formulas are frequently tested in CBSE Class 10 board exams:
Formula | Complementary Pair |
sin (90° - A) = cos A | sin and cos are complementary |
cos (90° - A) = sin A | cos and sin are complementary |
tan (90° - A) = cot A | tan and cot are complementary |
cot (90° - A) = tan A | cot and tan are complementary |
sec (90° - A) = cosec A | sec and cosec are complementary |
cosec (90° - A) = sec A | cosec and sec are complementary |
Key Point: Each trigonometric ratio has a "co-function" partner: sin-cos, tan-cot, sec-cosec. The prefix "co" literally stands for complementary. These formulas are essential for solving questions where the sum of two angles equals 90°.
Trigonometric Identities | Proof and Simplification
Beyond the Pythagorean identities, students are expected to prove and apply a variety of standard trigonometric identities. The following formulas appear regularly in CBSE Class 10 board exam questions (3-mark and 4-mark problems):
Useful Identity Expressions Derived from Pythagorean Identities
Expression | Simplified Value |
(sin A + cos A)² + (sin A - cos A)² | = 2 |
(sin A + cos A)² - (sin A - cos A)² | = 4 sin A cos A |
sin⁴A + cos⁴A | = 1 - 2sin²A cos²A |
sin⁶A + cos⁶A | = 1 - 3sin²A cos²A |
(1 + tan²A)(1 + cot²A) | = 1 / (sin²A - sin⁴A) |
tan A + cot A | = sec A cosec A |
Exam Strategy: When proving trigonometric identities, always start with the more complex side and simplify it to match the simpler side. Never move terms across the equal sign as you would in equations — treat each side independently.
Heights and Distances — Applied Trigonometry Formulas
Chapter 9 of CBSE Class 10 Maths covers the real-life applications of trigonometry through Heights and Distances. These problems use two special angles:
Angle | Definition |
Angle of Elevation | The angle formed between the horizontal line and the line of sight when looking UP at an object. Used when the observer is at a lower level than the object. |
Angle of Depression | The angle formed between the horizontal line and the line of sight when looking DOWN at an object. Used when the observer is at a higher level than the object. |
Key Formulas Used in Heights and Distances
Situation | Formula |
Height of object (tower, building, tree) | Height = Distance × tan(angle of elevation) |
Distance from the base | Distance = Height / tan(angle of elevation) |
Slant distance (line of sight) | Slant distance = Height / sin(angle of elevation) |
When two angles are from two different points | Use tan for both points and solve the system of equations |
Remember: The angle of elevation = angle of depression when the observer and the object are at different heights and the line connecting them creates parallel horizontal lines (alternate interior angles).
Summary: All Important Trigonometry Formulas at a Glance
Here is a quick-reference summary of all the key trigonometry formulas for Class 10:
• Six Ratios: sin = P/H, cos = B/H, tan = P/B, cosec = H/P, sec = H/B, cot = B/P
• Reciprocal: sin×cosec = 1, cos×sec = 1, tan×cot = 1
• Quotient: tan = sin/cos, cot = cos/sin
• Pythagorean Identity 1: sin²A + cos²A = 1
• Pythagorean Identity 2: 1 + tan²A = sec²A
• Pythagorean Identity 3: 1 + cot²A = cosec²A
• Complementary: sin(90°-A) = cosA, cos(90°-A) = sinA, tan(90°-A) = cotA
• Standard Angles: sin30° = 1/2, cos60° = 1/2, tan45° = 1, sin90° = 1
• Heights & Distances: Height = Distance × tan(elevation angle)
Tips to Memorize Trigonometry Formulas for Class 10
Memorizing formulas alone is not enough, you need to understand how to apply them. Here are proven strategies:
• Create a Formula Chart: Write all formulas on a single sheet and paste it at your study desk for daily revision.
• Practice Daily: Solve at least 5 trigonometry problems every day in the weeks leading up to your board exam.
• Use Mnemonics: "Some People Have Curly Black Hair" for sin, cos, tan ratios. The 0-1-2-3-4 pattern for standard angle values.
• Solve NCERT Thoroughly: All examples and exercise problems in NCERT Class 10 Maths Chapter 8 and 9 must be solved and understood.
• Attempt Previous Year Papers: CBSE board papers from the past 10 years contain repeated formula-based questions — practice them for pattern recognition.
• Understand Derivations: Don't just memorize, understand where each identity comes from. This helps reconstruct forgotten formulas during the exam.
Conclusion
Mastering all important trigonometry formulas for Class 10 is not just about passing your board exam, it is about building a strong mathematical foundation for your academic journey ahead. The formulas covered in this guide, from the six trigonometric ratios, reciprocal and quotient identities, to Pythagorean identities, standard angle values, complementary angle formulas, and Heights & Distances applications, form the complete trigonometry syllabus for Class 10.
Practice consistently, revise this formula sheet regularly, and solve a variety of NCERT and past paper problems. With the right preparation, scoring full marks in trigonometry in Class 10 is absolutely achievable. Good luck!
Frequently Asked Questions (FAQs)
Q1: How many trigonometric ratios are there in Class 10?
There are six trigonometric ratios in Class 10: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). They are defined based on the ratio of the sides of a right-angled triangle.
Q2: What are the most important trigonometry formulas for Class 10 boards?
The most important formulas for CBSE Class 10 board exams include the Pythagorean identities (sin²A + cos²A = 1, 1 + tan²A = sec²A, 1 + cot²A = cosec²A), the values of standard angles (0°, 30°, 45°, 60°, 90°), and complementary angle formulas.
Q3: What is the easiest way to remember trigonometry standard angle values?
The easiest method is the 0-1-2-3-4 trick: For sin values from 0° to 90°, write √0/2, √1/2, √2/2, √3/2, √4/2 and simplify. For cos, write the values in reverse order.
Q4: Are trigonometry formulas from Class 10 useful in Class 11 and 12?
Yes, absolutely. Class 10 trigonometry formulas are the foundation for Class 11 and 12 topics including compound angles, multiple angles, inverse trigonometry, and calculus. Mastering them now gives you a significant head start.
Q5: How many marks does trigonometry carry in CBSE Class 10 boards?
Trigonometry (Chapters 8 and 9) typically carries 12–14 marks in CBSE Class 10 board examinations out of 80 marks total, making it one of the highest-weightage topics in the paper.
