Essential Trigonometric Formulas You Need to Know
The Trig Formulas That Actually Matter
Most textbooks throw 50+ formulas at you. You need maybe 15. Here's what actually works.
The Basic Ratios (Your Foundation)
If you forget everything else, remember these three. Everything else in trig is built on them.
- Sin θ = Opposite / Hypotenuse
- Cos θ = Adjacent / Hypotenuse
- Tan θ = Opposite / Adjacent (which is also sin θ / cos θ)
That's it. The entire rest of trigonometry is just variations and combinations of these three relationships.
The Big Six Pythagorean Identities
These connect the three basic ratios. You'll use the first one constantly.
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
The first identity is the most important. Derive the others from it when you forget them.
Reciprocal Identities
Sometimes you need the flipped versions. Here they are:
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
Double Angle Formulas
When you have 2θ, use these:
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ
- Also: cos 2θ = 2cos²θ − 1 = 1 − 2sin²θ
- tan 2θ = 2 tan θ / (1 − tan²θ)
Sum and Difference Formulas
For adding or subtracting angles:
- sin(A ± B) = sin A cos B ± cos 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)
The plus/minus signs flip in the same positions. Watch where they go.
Half Angle Formulas
These come from the double angle formulas. Useful for integrals and solving equations.
- sin(θ/2) = ±√((1 − cos θ) / 2)
- cos(θ/2) = ±√((1 + cos θ) / 2)
- tan(θ/2) = (1 − cos θ) / sin θ = sin θ / (1 + cos θ)
The ± depends on which quadrant θ/2 lands in.
Product-to-Sum Formulas
These convert products into sums. Useful for integration and signal processing.
- sin A cos B = ½[sin(A + B) + sin(A − B)]
- cos A sin B = ½[sin(A + B) − sin(A − B)]
- cos A cos B = ½[cos(A + B) + cos(A − B)]
- sin A sin B = ½[cos(A − B) − cos(A + B)]
Quick Reference Table
| Formula Type | Key Formula |
|---|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| Double Angle (sin) | sin 2θ = 2 sin θ cos θ |
| Double Angle (cos) | cos 2θ = cos²θ − sin²θ |
| Sum (sin) | sin(A + B) = sin A cos B + cos A sin B |
| Sum (cos) | cos(A + B) = cos A cos B − sin A sin B |
| Tangent Sum | tan(A + B) = (tan A + tan B) / (1 − tan A tan B) |
| Reciprocal | csc θ = 1 / sin θ |
How to Actually Use These
Stop memorizing everything. Use this approach:
- Know the three basic ratios cold. Draw a right triangle if you have to.
- Derive identities from sin²θ + cos²θ = 1 when stuck. Move terms, divide, substitute.
- For sin 2θ, think "2 times sin times cos." For cos 2θ, think "cos² minus sin²."
- For tan, use sin/cos. tan(A + B) = (sin A cos B + cos A sin B) / (cos A cos B − sin A sin B), then divide numerator and denominator by cos A cos B.
The Pattern You Might Have Missed
Most trig formulas follow predictable patterns. The sum and difference formulas for sine and cosine look almost identical with small sign changes. Once you see the symmetry, you only need to memorize one and derive the rest.
Same with Pythagorean identities. Start with sin²θ + cos²θ = 1. Divide by cos²θ to get tan²θ + 1 = sec²θ. Divide by sin²θ to get 1 + cot²θ = csc²θ.
You don't need to memorize everything. You need to understand the relationships.