Complete Trigonometry Formulas: All Important Formulas for Class 10, 11 & 12
Why it matters:
Every competitive exam (JEE, NEET, SAT) tests trigonometry heavily. Mastering these formulas gives you a solid foundation.
How to use it:
Understand the basic ratios first, then move to identities and transformations.
Basic Trigonometry Ratios
These are the foundation of all trigonometry. In a right-angled triangle:
| Ratio | Formula | Explanation | Variables |
|---|---|---|---|
| Sine | sin θ = Opposite/Hypotenuse | Ratio of opposite side to hypotenuse | θ = angle |
| Cosine | cos θ = Adjacent/Hypotenuse | Ratio of adjacent side to hypotenuse | θ = angle |
| Tangent | tan θ = Opposite/Adjacent | Ratio of opposite to adjacent side | θ = angle |
| Cosecant | cosec θ = Hypotenuse/Opposite | Reciprocal of sine | θ = angle |
| Secant | sec θ = Hypotenuse/Adjacent | Reciprocal of cosine | θ = angle |
| Cotangent | cot θ = Adjacent/Opposite | Reciprocal of tangent | θ = angle |
Memory Trick: SOHCAHTOA
- Sin = Opposite/Hypotenuse
- Cos = Adjacent/Hypotenuse
- Tan = Opposite/Adjacent
Trigonometry Table (Standard Angles)
This table is gold for exams. Memorize it.
| Angle (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 | ∞ |
| cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | ∞ |
| cot θ | ∞ | √3 | 1 | 1/√3 | 0 |
Pattern for sin θ: √0/2, √1/2, √2/2, √3/2, √4/2
Pattern for cos θ: Reverse of sin (√4/2, √3/2, √2/2, √1/2, √0/2)
Reciprocal Identities
| Formula Name | Formula | Explanation |
|---|---|---|
| Cosecant | cosec θ = 1/sin θ | Reciprocal of sine |
| Secant | sec θ = 1/cos θ | Reciprocal of cosine |
| Cotangent | cot θ = 1/tan θ | Reciprocal of tangent |
| Alternative tan | tan θ = sin θ/cos θ | Tangent as ratio of sin and cos |
| Alternative cot | cot θ = cos θ/sin θ | Cotangent as ratio of cos and sin |
Pythagorean Identities
These come directly from the Pythagorean theorem.
| Formula | Explanation | Use Case |
|---|---|---|
| sin²θ + cos²θ = 1 | Most fundamental identity | Simplifying expressions |
| 1 + tan²θ = sec²θ | Derived by dividing first identity by cos²θ | Solving sec, tan problems |
| 1 + cot²θ = cosec²θ | Derived by dividing first identity by sin²θ | Solving cosec, cot problems |
Example: If sin θ = 3/5, find cos θ.
Using sin²θ + cos²θ = 1:
- (3/5)² + cos²θ = 1
- 9/25 + cos²θ = 1
- cos²θ = 16/25
- cos θ = 4/5
Trigonometric Ratios of Complementary Angles
Complementary angles add up to 90°.
| Formula | Explanation |
|---|---|
| sin(90° - θ) = cos θ | Sine becomes cosine |
| cos(90° - θ) = sin θ | Cosine becomes sine |
| tan(90° - θ) = cot θ | Tangent becomes cotangent |
| cot(90° - θ) = tan θ | Cotangent becomes tangent |
| sec(90° - θ) = cosec θ | Secant becomes cosecant |
| cosec(90° - θ) = sec θ | Cosecant becomes secant |
Memory Trick: Co-function of (90° - θ) = function of θ
Sum and Difference Formulas
These help you find trigonometric values of sum or difference of two angles.
| Formula Name | Formula |
|---|---|
| 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) |
Example: Find sin 75°.
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (1/√2)(√3/2) + (1/√2)(1/2) = (√3 + 1)/2√2
Double Angle Formulas
Used when angle is doubled.
| Formula Name | Formula | Alternative Forms |
|---|---|---|
| sin 2θ | 2 sin θ cos θ | — |
| cos 2θ | cos²θ - sin²θ | 2cos²θ - 1 or 1 - 2sin²θ |
| tan 2θ | 2tan θ/(1 - tan²θ) | — |
Example: If sin θ = 3/5, find sin 2θ.
sin 2θ = 2 sin θ cos θ = 2(3/5)(4/5) = 24/25
Triple Angle Formulas
For Class 11 and 12 advanced problems.
| Formula | Expression |
|---|---|
| sin 3θ | 3sin θ - 4sin³θ |
| cos 3θ | 4cos³θ - 3cos θ |
| tan 3θ | (3tan θ - tan³θ)/(1 - 3tan²θ) |
Half Angle Formulas
| Formula | Expression |
|---|---|
| sin θ/2 | ±√[(1 - cos θ)/2] |
| cos θ/2 | ±√[(1 + cos θ)/2] |
| tan θ/2 | ±√[(1 - cos θ)/(1 + cos θ)] or sin θ/(1 + cos θ) |
The sign depends on which quadrant θ/2 lies in.
Product-to-Sum Formulas
Convert products into sums for easier integration and simplification.
| Formula | Expression |
|---|---|
| sin A sin B | [cos(A - B) - cos(A + B)]/2 |
| cos A cos B | [cos(A - B) + cos(A + B)]/2 |
| sin A cos B | [sin(A + B) + sin(A - B)]/2 |
| cos A sin B | [sin(A + B) - sin(A - B)]/2 |
Sum-to-Product Formulas
Convert sums into products.
| Formula | Expression |
|---|---|
| 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] |
Inverse Trigonometry Formulas
Basic Inverse Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| sin⁻¹x | [-1, 1] | [-π/2, π/2] |
| cos⁻¹x | [-1, 1] | [0, π] |
| tan⁻¹x | (-∞, ∞) | (-π/2, π/2) |
| cosec⁻¹x | (-∞, -1] ∪ [1, ∞) | [-π/2, 0) ∪ (0, π/2] |
| sec⁻¹x | (-∞, -1] ∪ [1, ∞) | [0, π/2) ∪ (π/2, π] |
| cot⁻¹x | (-∞, ∞) | (0, π) |
Important Inverse Trigonometry Properties
| Property | Formula |
|---|---|
| sin⁻¹(-x) | -sin⁻¹x |
| cos⁻¹(-x) | π - cos⁻¹x |
| tan⁻¹(-x) | -tan⁻¹x |
| sin⁻¹x + cos⁻¹x | π/2 |
| tan⁻¹x + cot⁻¹x | π/2 |
| sec⁻¹x + cosec⁻¹x | π/2 |
| tan⁻¹x + tan⁻¹y | tan⁻¹[(x + y)/(1 - xy)] |
| tan⁻¹x - tan⁻¹y | tan⁻¹[(x - y)/(1 + xy)] |
| 2tan⁻¹x | sin⁻¹[2x/(1 + x²)] or cos⁻¹[(1 - x²)/(1 + x²)] |
Solved Examples
Example 1: Using Pythagorean Identity
Problem: If tan θ = 3/4, find sec θ.
Solution: Using 1 + tan²θ = sec²θ
- 1 + (3/4)² = sec²θ
- 1 + 9/16 = sec²θ
- 25/16 = sec²θ
- sec θ = 5/4
Example 2: Sum Formula Application
Problem: Find the value of sin 15°.
Solution: sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (1/√2)(√3/2) - (1/√2)(1/2) = (√3 - 1)/2√2
Example 3: Inverse Trigonometry
Problem: Simplify tan⁻¹(1) + cos⁻¹(1/2)
Solution: tan⁻¹(1) = π/4 (since tan π/4 = 1) cos⁻¹(1/2) = π/3 (since cos π/3 = 1/2) Answer = π/4 + π/3 = 7π/12
Common Mistakes to Avoid
Mistake 1: Confusing sin²θ with sin(θ²)
- sin²θ means (sin θ)², not sin of θ squared
Mistake 2: Forgetting domain restrictions in inverse functions
- sin⁻¹(2) is undefined because sin x can never equal 2
Mistake 3: Using degree mode when answer needs radians (or vice versa)
- Always check calculator settings
Mistake 4: Mixing up complementary angle formulas
- sin(90° - θ) = cos θ, not sin θ
Mistake 5: Wrong signs in sum/difference formulas
- cos(A + B) has a minus: cos A cos B - sin A sin B
Tips & Memory Tricks for Students
For Standard Angles:
- Use the √0, √1, √2, √3, √4 pattern divided by 2 for sin
- Reverse it for cos
For Complementary Angles:
- "Co" functions swap at 90°
- sin ↔ cos, tan ↔ cot, sec ↔ cosec
For Sum Formulas:
- sin keeps order: sin A cos B + cos A sin B
- cos swaps and changes sign: cos A cos B - sin A sin B
For Pythagorean Identities:
- Remember sin²θ + cos²θ = 1 first
- Divide by cos²θ to get tan/sec form
- Divide by sin²θ to get cot/cosec form
Frequently Asked Questions about Trigonometry Formula
Q. What are the 6 basic trigonometry formulas?
The six basic ratios are sin θ, cos θ, tan θ, cosec θ, sec θ, and cot θ. Use SOHCAHTOA to remember: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. The other three are their reciprocals.
Q. How do I memorize the trigonometry table quickly?
Use the √n/2 pattern for sin values (n = 0,1,2,3,4), then reverse for cos. Write it 5 times daily for 3 days. Test yourself without looking. Repetition builds memory faster than staring at the table.
Q. What's the difference between sin⁻¹x and 1/sin x?
sin⁻¹x (or arcsin x) is the inverse function that gives you an angle. 1/sin x is cosec x, the reciprocal ratio. They're completely different. sin⁻¹(1/2) = 30°, but 1/sin(30°) = 2.
Q. Which trigonometry formulas are most important for Class 10?
Focus on basic ratios, standard angle values, Pythagorean identities (sin²θ + cos²θ = 1), and complementary angle formulas. Class 10 rarely goes beyond these basics, so master them thoroughly before moving on.
Q. How do I know which formula to use in a problem?
Look at what's given and what's asked. If you have one function and need another, use identities. If you have two angles, use sum formulas. Practice identifying patterns in problems.
Q. Are inverse trigonometry formulas hard to learn?
Not if you know their domains and ranges first. Remember that sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2. These two properties solve 60% of inverse trig problems in Class 12 exams.
Q. Can I use these formulas in competitive exams like JEE?
Absolutely. JEE tests trigonometry heavily. You need all these formulas plus their applications in calculus and coordinate geometry. Make a formula sheet and revise it weekly. Speed and accuracy both matter in JEE.
Q. What's the best way to practice trigonometry?
Solve 5 problems daily using different formula types. Mix basic problems with challenging ones. Time yourself. Teach concepts to a friend. Active problem-solving builds mastery faster than passive reading ever will.
Trigonometry isn't about memorizing random formulas. It's about understanding patterns and relationships. Once you see how each formula connects to the others how Pythagorean identities lead to double angles, how complementary angles simplify complex problems it all starts making sense.