Trigonometry Formulas

Trigonometry Formulas (Class 10, 11 & 12 Master List)

Trigonometry formulas are mathematical expressions that define the relationships between the angles and sides of a right-angled triangle. This comprehensive formula list serves as a master reference sheet spanning foundational concepts in CBSE Class 10, advanced algebraic identities in Class 11, and calculus applications for Class 12, JEE Main, and NEET exams.

Class: 10, 11 & 12
Topic: Trigonometric Ratios & Identities
Exams: CBSE · ICSE · JEE · NDA

1. Foundational Class 10 Trigonometry Formulas

Trigonometry begins with the right-angled triangle, comparing the lengths of its three sides: Perpendicular (P), Base (B), and Hypotenuse (H).

Trigonometric Ratio Right-Triangle Ratio Reciprocal Relation
sin θ (Sine) Perpendicular / Hypotenuse (P/H) 1 / cosec θ
cos θ (Cosine) Base / Hypotenuse (B/H) 1 / sec θ
tan θ (Tangent) Perpendicular / Base (P/B) 1 / cot θ  [sin θ / cos θ]
cosec θ (Cosecant) Hypotenuse / Perpendicular (H/P) 1 / sin θ
sec θ (Secant) Hypotenuse / Base (H/B) 1 / cos θ
cot θ (Cotangent) Base / Perpendicular (B/P) 1 / tan θ  [cos θ / sin θ]

Three Main Pythagorean Identities

Derived via the Pythagoras theorem ($P^2 + B^2 = H^2$), these equations are essential for substituting variables:

  • sin2θ + cos2θ = 1  ⇒  sin2θ = 1 − cos2θ
  • 1 + tan2θ = sec2θ  ⇒  sec2θ − tan2θ = 1
  • 1 + cot2θ = cosec2θ  ⇒  cosec2θ − cot2θ = 1

2. Trigonometric Standard Values Table (0° to 90°)

Memorizing these primary values helps students fast-track through tricky algebraic geometry evaluations across Class 10 and 11 exercises.

Ratio 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

3. ASTC Sign Rule (All Quadrants)

In Class 11, angles expand beyond 90°. The **ASTC rule** ("All Silver Tea Cups") dictates which identities maintain positive outputs across the 4 coordinate zones:

Quadrant I (0° to 90°): **A**ll functions are Positive.
Quadrant II (90° to 180°): **S**in and Cosec are Positive. Others are negative.
Quadrant III (180° to 270°): **T**an and Cot are Positive. Others are negative.
Quadrant IV (270° to 360°): **C**os and Sec are Positive. Others are negative.

4. Class 11 Advanced Trigonometric Identities

These compound and multiple angle systems are critical for solving advanced functions in Calculus and Coordinate Geometry challenges.

Compound Angle Ratios

  • 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

Double Angle Formulas (Very Important)

sin 2x Formulas:
sin 2x = 2 sin x cos x = 2 tan x1 + tan2x

cos 2x Formulas:
cos 2x = cos2x − sin2x = 2cos2x − 1 = 1 − 2sin2x

5. Product-to-Sum & Sum-to-Product Formulas

Transforming multi-variable transformations from multiplications to single addition parameters simplifies integral systems.

2 sin x cos y Formula: sin(x + y) + sin(x − y)
2 cos x sin y Formula: sin(x + y) − sin(x − y)
2 cos x cos y Formula: cos(x + y) + cos(x − y)
2 sin x sin y Formula: cos(x − y) − cos(x + y)

6. Comprehensive Solved Examples

Review these multi-level application examples to master problem-solving mechanics across schools and entry competitive tiers.

Example 1: Double-Angle Simplification (Class 10/11)

Evaluate the expression without using a calculator: 2 sin(15°) cos(15°).

Solution:
Recognize that this expression matches the double-angle template: 2 sin θ cos θ = sin 2θ.
Assigning θ = 15°:
2 sin(15°) cos(15°) = sin(2 × 15°) = sin(30°).
From our standard values table, sin(30°) = 1/2.
Answer: 1/2 (or 0.5)
Example 2: Pythagorean Trigonometric Identity Transformation

If sin θ = 3/5 and θ lies in the second quadrant, determine the value of cos θ and tan θ.

Solution:
Use the identity: sin2θ + cos2θ = 1.
(3/5)2 + cos2θ = 1 ⇒ 9/25 + cos2θ = 1
cos2θ = 1 − 9/25 = 16/25
cos θ = ± 4/5.

Applying the ASTC Rule: In the second quadrant, Cosine is negative. Therefore, cos θ = −4/5.
Now, compute Tangent using the quotient relationship:
tan θ = sin θ / cos θ = (3/5) / (−4/5) = −3/4.
Answer: cos θ = −4/5, tan θ = −3/4

Example 3: Compound Angle Expansion (Class 11)

Determine the exact numerical value of cos(75°).

Solution:
Express 75° as the sum of standard angles: 75° = 45° + 30°.
Apply the identity: cos(A + B) = cos A cos B − sin A sin B.
cos(45° + 30°) = cos(45°)cos(30°) − sin(45°)sin(30°)

Substitute the values from the reference chart:
= (1/√2 × √3/2) − (1/√2 × 1/2)
= √3 / 2√2 − 1 / 2√2 = (√3 − 1) / 2√2.
Answer: (√3 − 1) / 2√2

Example 4: Product-to-Sum Identity Application (Teachoo Style)

Express the product term 2 sin(5x) sin(3x) as a pure sum or difference expression.

Solution:
Recall the targeted transformation identity: 2 sin x sin y = cos(x − y) − cos(x + y).
Here, substitute x = 5x and y = 3x into the identity fields:
= cos(5x − 3x) − cos(5x + 3x)
= cos(2x) − cos(8x)
Answer: cos(2x) − cos(8x)
Example 5: Multi-Angle Equation Form (JEE Level)

Prove or evaluate the expression value: (1 − cos 2A) / sin 2A.

Solution:
Substitute the basic multiple angle components:
We know that 1 − cos 2A = 2 sin2A and sin 2A = 2 sin A cos A.

Placing these into the fraction:
= (2 sin2A) / (2 sin A cos A)
Cancel out common components (2 and sin A):
= sin A / cos A = tan A.
Answer: tan A

7. Frequently Asked Questions (FAQs)

What are the fundamental formulas in trigonometry?
The fundamental ratios are based on a right triangle: sin θ = P/H, cos θ = B/H, and tan θ = P/B. These values are tied together by the core Pythagorean identity: sin2θ + cos2θ = 1.
How do I remember the sign of ratios across quadrants?
Use the mnemonic "All Silver Tea Cups" (ASTC). It maps to Quadrants I, II, III, and IV respectively, showcasing that All functions, Sine, Tan, or Cosine are the unique positive elements in those zones.
What is the formula expansion for 2 sin x sin y?
The product-to-sum identity states that 2 sin x sin y = cos(x − y) − cos(x + y). This conversion is widely useful in Class 12 integral calculus tasks.

FAQs (Frequently Asked Questions)

Trigonometry is a branch of mathematics that specifically deals with the angles of a triangle and tries to see the relationship between each element of the triangle which is three sides and three angles

Try to get the actual relationship of each formula at least once, and then go for memorising each formula every day, try to understand the relationship and then we can go for analysing and solving for the same.

 

Sin A = Perpendicular/Hypotenuse
Cos A = Base/Hypotenuse
Tan A = Perpendicular/Base

The formula for sin 3x is 3sin x – 4sin3x.