Trigonometric Identities
Every standard trig identity in one organized reference — and an interactive unit circle that shows sin θ, cos θ and tan θ live as you drag the angle, so the formulas actually make sense.
Drag the slider to change the angle and watch sin, cos and tan update in real time on the unit circle.
All trigonometric identities
Pythagorean
- sin²θ + cos²θ=1
- 1 + tan²θ=sec²θ
- 1 + cot²θ=csc²θ
Reciprocal
- cscθ=1 / sinθ
- secθ=1 / cosθ
- cotθ=1 / tanθ
Quotient
- tanθ=sinθ / cosθ
- cotθ=cosθ / sinθ
Even-Odd
- sin(−θ)=−sinθ
- cos(−θ)=cosθ
- tan(−θ)=−tanθ
Cofunction
- sin(90° − θ)=cosθ
- cos(90° − θ)=sinθ
- tan(90° − θ)=cotθ
Sum & Difference
- sin(α ± β)=sinα·cosβ ± cosα·sinβ
- cos(α ± β)=cosα·cosβ ∓ sinα·sinβ
- tan(α ± β)=(tanα ± tanβ) / (1 ∓ tanα·tanβ)
Double-Angle
- sin2θ=2·sinθ·cosθ
- cos2θ=cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
- tan2θ=2tanθ / (1 − tan²θ)
Half-Angle
- sin(θ/2)=±√((1 − cosθ) / 2)
- cos(θ/2)=±√((1 + cosθ) / 2)
- tan(θ/2)=(1 − cosθ) / sinθ
Product-to-Sum
- sinα·cosβ=½[sin(α + β) + sin(α − β)]
- cosα·cosβ=½[cos(α − β) + cos(α + β)]
- sinα·sinβ=½[cos(α − β) − cos(α + β)]
Law of Sines & Cosines
- a / sinA = b / sinB=c / sinC
- c²=a² + b² − 2ab·cosC
All trigonometric identities
Pythagorean
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Reciprocal
- cscθ = 1 / sinθ
- secθ = 1 / cosθ
- cotθ = 1 / tanθ
Quotient
- tanθ = sinθ / cosθ
- cotθ = cosθ / sinθ
Even-Odd
- sin(−θ) = −sinθ
- cos(−θ) = cosθ
- tan(−θ) = −tanθ
Cofunction
- sin(90° − θ) = cosθ
- cos(90° − θ) = sinθ
- tan(90° − θ) = cotθ
Sum & Difference
- sin(α ± β) = sinα·cosβ ± cosα·sinβ
- cos(α ± β) = cosα·cosβ ∓ sinα·sinβ
- tan(α ± β) = (tanα ± tanβ) / (1 ∓ tanα·tanβ)
Double-Angle
- sin2θ = 2·sinθ·cosθ
- cos2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
- tan2θ = 2tanθ / (1 − tan²θ)
Half-Angle
- sin(θ/2) = ±√((1 − cosθ) / 2)
- cos(θ/2) = ±√((1 + cosθ) / 2)
- tan(θ/2) = (1 − cosθ) / sinθ
Product-to-Sum
- sinα·cosβ = ½[sin(α + β) + sin(α − β)]
- cosα·cosβ = ½[cos(α − β) + cos(α + β)]
- sinα·sinβ = ½[cos(α − β) − cos(α + β)]
Law of Sines & Cosines
- a / sinA = b / sinB = c / sinC
- c² = a² + b² − 2ab·cosC
About the Trigonometric Identities reference
Trigonometric identities are equations that are true for every value of the angle, and they are the backbone of trigonometry, precalculus and calculus. This reference collects all of the ones students actually need, grouped the way teachers introduce them: the Pythagorean identities (sin²θ + cos²θ = 1 and its two siblings), the reciprocal and quotient identities that define csc, sec, cot and tan, the even-odd identities that describe symmetry, the cofunction identities that link sine to cosine, the sum and difference formulas, the double-angle and half-angle formulas, the product-to-sum formulas, and the law of sines and law of cosines for solving any triangle. What sets this page apart from a flat table is the interactive unit circle at the top: drag the angle and watch the point move while sin θ, cos θ and tan θ update live, so you can see why, for example, sin and cos are bounded by 1 and why tan blows up to infinity near 90°. Every formula is written in clean Unicode notation — no images, so it copies cleanly and is easy for assistive tech and AI to read. Use it as a homework cheat sheet, an exam revision list, or a quick lookup when you are simplifying an expression or proving an identity.
How to use this reference
- 1 Drag the angle slider (or turn on snap-to-special-angles) and watch the point on the unit circle move.
- 2 Read the live sin θ, cos θ and tan θ values to build intuition for how the functions behave around the circle.
- 3 Scroll the identity cards below — grouped by type — to find the formula you need and copy it into your work.
How the identities relate to the unit circle
On a unit circle of radius 1, a point at angle θ measured counter-clockwise from the positive x-axis has coordinates (cos θ, sin θ). That single picture explains most of the identities: • Because the point lies on a circle of radius 1, x² + y² = 1 gives the fundamental Pythagorean identity sin²θ + cos²θ = 1; dividing it by cos²θ or sin²θ produces 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. • tan θ = sin θ / cos θ is the slope of the radius, which is why it grows without bound as cos θ approaches 0 near 90° and 270° (shown as "undefined"). • Reflecting the point across the x-axis sends θ to −θ and flips the sign of y but not x, giving the even-odd identities sin(−θ) = −sin θ and cos(−θ) = cos θ. • Swapping x and y (a reflection across the line y = x) maps θ to 90° − θ, which is the cofunction relationship sin(90° − θ) = cos θ. The sum, difference, double-angle, half-angle and product-to-sum formulas are then derived algebraically from these, and the laws of sines and cosines extend the same trigonometry to triangles that are not right-angled.
Frequently asked questions
What are the basic trigonometric identities?
The most fundamental are the Pythagorean identities — sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ — together with the reciprocal identities (csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ) and the quotient identities (tan θ = sin θ/cos θ, cot θ = cos θ/sin θ). Almost every other identity can be derived from these, which is why they are usually the first ones memorized.
What is the Pythagorean identity?
The Pythagorean identity is sin²θ + cos²θ = 1, true for every angle θ. It comes straight from the unit circle: a point at angle θ has coordinates (cos θ, sin θ), and because the circle has radius 1, the Pythagorean theorem gives cos²θ + sin²θ = 1. Dividing both sides by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.
What are the double-angle formulas?
The double-angle formulas express trig functions of 2θ in terms of θ: sin 2θ = 2·sin θ·cos θ; cos 2θ = cos²θ − sin²θ, which can also be written 1 − 2sin²θ or 2cos²θ − 1; and tan 2θ = 2tan θ / (1 − tan²θ). They are special cases of the sum formulas with α = β = θ, and they are used constantly when simplifying expressions and solving trigonometric equations.
When do I use the law of sines versus the law of cosines?
Use the law of sines, a/sin A = b/sin B = c/sin C, when you know two angles and one side, or two sides and a non-included angle. Use the law of cosines, c² = a² + b² − 2ab·cos C, when you know two sides and the angle between them (to find the third side) or all three sides (to find an angle). Both let you solve triangles that are not right-angled, where the basic sin/cos/tan ratios alone are not enough.
Related tools
Keep exploring trigonometry on UnitConv: drag the full interactive unit circle to see all six trig functions with exact values, plot sine and cosine waves with the graphing calculator, and tidy up large or small numbers with the scientific notation converter.