Sine, cosine and tangent
- sin θ = a / c
- cos θ = b / c
- tan θ = a / b
a opposite sideb adjacent sidec hypotenuse
Trigonometry Formulas
Every trigonometry formula on one sheet: the three ratios, the identities that follow from them, the angle formulas, the laws of sines and cosines, radians and graphs.
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Every trigonometry formula on one page: the three ratios, the identities, the angle formulas, the laws for any triangle, radians and graphs. Each symbol is named under its formula. Switch to quiz mode to hide the formulas and test yourself, or print the whole sheet.
Trig ratios
Reciprocal identities
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
Quotient identities
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Special angles
Anglesincostan
0°010
30°π/61/2√3/2√3/3
45°π/4√2/2√2/21
60°π/3√3/21/2√3
90°π/210—
Identities
Pythagorean identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Even and odd identities
- sin(−θ) = −sin θ
- cos(−θ) = cos θ
- tan(−θ) = −tan θ
Cofunction identities
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- tan(90° − θ) = cot θ
Angle formulas
Sum and difference formulas
- 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)
Double angle formulas
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ
- cos 2θ = 2cos²θ − 1 = 1 − 2sin²θ
- tan 2θ = 2 tan θ / (1 − tan²θ)
Half angle formulas
- 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)
Power reducing formulas
- sin²θ = (1 − cos 2θ) / 2
- cos²θ = (1 + cos 2θ) / 2
Product to sum formulas
- sin A sin B = ½[cos(A − B) − cos(A + B)]
- cos A cos B = ½[cos(A − B) + cos(A + B)]
- sin A cos B = ½[sin(A + B) + sin(A − B)]
Sum to product formulas
- 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)
Triangles
Law of sines
- a / sin A = b / sin B = c / sin C
A, B, C anglesa, b, c the sides opposite them
Law of cosines
- c² = a² + b² − 2ab cos C
A, B, C anglesa, b, c the sides opposite them
Area of a triangle
- Area
- A = ½ab sin C
C the angle between sides a and b
Radians and graphs
Radians, arcs and sectors
- 180° = π rad
- Degrees to radians
- θ(rad) = θ(°) · π / 180°
- Arc length
- s = rθ
- Sector area
- A = ½r²θ
r radiusθ angle in radianss arc length
Sine and cosine graphs
- y = A sin(Bx − C) + D
- Amplitude
- |A|
- Period
- 2π / |B|
- Phase shift
- C / B
- Midline
- y = D
Everything starts with three ratios
In a right triangle, pick one of the two acute angles and call it θ. The side across from it is the opposite side, the side next to it is the adjacent side, and the longest side is the hypotenuse. Then:
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- tan θ = opposite / adjacent
Almost every other line on the sheet is one of these three, rearranged.
The identities are one fact, divided
Put the triangle inside a circle of radius 1. The hypotenuse is 1, so the two legs are exactly cos θ and sin θ, and the Pythagorean theorem gives sin²θ + cos²θ = 1.
That one line gives the other two. Divide every term by cos²θ and you get tan²θ + 1 = sec²θ; divide by sin²θ instead and you get 1 + cot²θ = csc²θ. The unit circle page shows the triangle moving round the circle.
The angle formulas come from two
Learn the sum formulas for sine and cosine. Everything in the angle formulas section follows from them:
- Double angle: set B = A in the sum formula, and sin(A + A) becomes 2 sin A cos A.
- Power reducing: solve cos 2θ = 1 − 2sin²θ for sin²θ.
- Half angle: the power reducing formula with θ written as half of 2θ, then a square root.
- Difference: replace B with −B and use the even and odd identities.
Any triangle, not just right ones
The ratios need a right angle. For any other triangle, label each angle with a capital letter and the side across from it with the same small letter. Then:
- The law of sines links each side to the angle opposite it. Use it when you know a side and its opposite angle.
- The law of cosines is the Pythagorean theorem with a correction for the angle: when C = 90°, cos C = 0 and it becomes c² = a² + b². Use it when you know two sides and the angle between them, or all three sides.
Degrees and radians
A full turn is 360°, or 2π radians, so 180° = π radians. The identities hold in either unit. The arc length s = rθ and the sector area A = ½r²θ need θ in radians: that is what makes them so short.
Try one
Where to go next
For where these values come from, read SOHCAHTOA and the unit circle. The geometry formula sheet has the area, volume and coordinate formulas.
Common questions
Is there a printable trig identities sheet?
Yes. Everything on this page is also a free letter-size PDF and a PNG. The download buttons are above the sheet and again under it.
What are the three Pythagorean identities?
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Only the first is worth memorizing: divide it by cos²θ to get the second and by sin²θ to get the third.
When do I use the law of sines and when the law of cosines?
Use the law of sines when you know an angle and the side opposite it, plus one more side or angle. Use the law of cosines when you know two sides and the angle between them, or all three sides. In a right triangle you need neither: the ratios sin, cos and tan are enough.
Which trig formulas should I memorize?
The three ratios, sin²θ + cos²θ = 1 and the sum formulas for sin and cos. The double angle formulas are the sum formulas with B = A, the half angle and power reducing formulas come from the double angle ones, and the reciprocal and quotient identities are definitions.
Do the formulas work in degrees and in radians?
Yes, every identity holds in either unit. Only the formulas that mix an angle with a length need radians: the arc length s = rθ and the sector area A = ½r²θ are true only when θ is in radians.
