Trig identities

Definitions

In a right triangle, for the angle θ.

• sin θ = oppositehypotenusesin of theta equals (opposite) over (hypotenuse)
• cos θ = adjacenthypotenusecos of theta equals (adjacent) over (hypotenuse)
• tan θ = oppositeadjacenttan of theta equals (opposite) over (adjacent)
• csc θ = hypotenuseoppositecsc of theta equals (hypotenuse) over (opposite)
• sec θ = hypotenuseadjacentsec of theta equals (hypotenuse) over (adjacent)
• cot θ = adjacentoppositecot of theta equals (adjacent) over (opposite)

Reciprocal and quotient

• csc θ = 1sin θcsc of theta equals (1) over (sin of theta)
• sec θ = 1cos θsec of theta equals (1) over (cos of theta)
• cot θ = 1tan θcot of theta equals (1) over (tan of theta)
• tan θ = sin θcos θtan of theta equals (sin of theta) over (cos of theta)
• cot θ = cos θsin θcot of theta equals (cos of theta) over (sin of theta)

Pythagorean

• sin² θ + cos² θ = 1sin squared of theta plus cos squared of theta equals 1
• 1 + tan² θ = sec² θ1 plus tan squared of theta equals sec squared of theta
• 1 + cot² θ = csc² θ1 plus cot squared of theta equals csc squared of theta

Even and odd (negative angle)

cos and sec are even; the other four are odd.

• sin(−θ) = −sin θsin of minus theta equals minus sin of theta
• cos(−θ) = cos θcos of minus theta equals cos of theta
• tan(−θ) = −tan θtan of minus theta equals minus tan of theta
• csc(−θ) = −csc θcsc of minus theta equals minus csc of theta
• sec(−θ) = sec θsec of minus theta equals sec of theta
• cot(−θ) = −cot θcot of minus theta equals minus cot of theta

Cofunction

π/2 is 90°, so in degrees sin(90° − θ) = cos θ.

• sin(π2 − θ) = cos θsin of (pi) over (2) minus theta equals cos of theta
• cos(π2 − θ) = sin θcos of (pi) over (2) minus theta equals sin of theta
• tan(π2 − θ) = cot θtan of (pi) over (2) minus theta equals cot of theta
• cot(π2 − θ) = tan θcot of (pi) over (2) minus theta equals tan of theta
• sec(π2 − θ) = csc θsec of (pi) over (2) minus theta equals csc of theta
• csc(π2 − θ) = sec θcsc of (pi) over (2) minus theta equals sec of theta

Periodicity

For any whole number n. sin, cos, csc and sec repeat every 2π (360°); tan and cot every π (180°).

• sin(θ + 2πn) = sin θsin of theta plus 2 pi n equals sin of theta
• cos(θ + 2πn) = cos θcos of theta plus 2 pi n equals cos of theta
• tan(θ + πn) = tan θtan of theta plus pi n equals tan of theta
• csc(θ + 2πn) = csc θcsc of theta plus 2 pi n equals csc of theta
• sec(θ + 2πn) = sec θsec of theta plus 2 pi n equals sec of theta
• cot(θ + πn) = cot θcot of theta plus pi n equals cot of theta

Sum and difference

Take the top signs together or the bottom signs together.

• sin(α ± β) = sin α cos β ± cos α sin βsin of alpha plus or minus beta equals sin of alpha cos of beta plus or minus cos of alpha sin of beta
• cos(α ± β) = cos α cos β ∓ sin α sin βcos of alpha plus or minus beta equals cos of alpha cos of beta minus or plus sin of alpha sin of beta
• tan(α ± β) = tan α ± tan β1 ∓ tan α tan βtan of alpha plus or minus beta equals (tan of alpha plus or minus tan of beta) over (1 minus or plus tan of alpha tan of beta)

Double angle

• sin 2θ = 2 sin θ cos θsin of 2 theta equals 2 sin of theta cos of theta
• cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θcos of 2 theta equals cos squared of theta minus sin squared of theta equals 2 cos squared of theta minus 1 equals 1 minus 2 sin squared of theta
• tan 2θ = 2 tan θ1 − tan² θtan of 2 theta equals (2 tan of theta) over (1 minus tan squared of theta)

Half angle

The sign of ± is the sign of the left side, which depends on the quadrant θ/2 is in.

• sin(θ2) = ±√1 − cos θ2sin of (theta) over (2) equals plus or minus the square root of ((1 minus cos of theta) over (2))
• cos(θ2) = ±√1 + cos θ2cos of (theta) over (2) equals plus or minus the square root of ((1 plus cos of theta) over (2))
• tan(θ2) = 1 − cos θsin θ = sin θ1 + cos θtan of (theta) over (2) equals (1 minus cos of theta) over (sin of theta) equals (sin of theta) over (1 plus cos of theta)

Power reducing

• sin² θ = 1 − cos 2θ2sin squared of theta equals (1 minus cos of 2 theta) over (2)
• cos² θ = 1 + cos 2θ2cos squared of theta equals (1 plus cos of 2 theta) over (2)
• tan² θ = 1 − cos 2θ1 + cos 2θtan squared of theta equals (1 minus cos of 2 theta) over (1 plus cos of 2 theta)

Product to sum

• sin α sin β = 12(cos(α − β) − cos(α + β))sin of alpha sin of beta equals (1) over (2) (cos of alpha minus beta minus cos of alpha plus beta)
• cos α cos β = 12(cos(α − β) + cos(α + β))cos of alpha cos of beta equals (1) over (2) (cos of alpha minus beta plus cos of alpha plus beta)
• sin α cos β = 12(sin(α + β) + sin(α − β))sin of alpha cos of beta equals (1) over (2) (sin of alpha plus beta plus sin of alpha minus beta)
• cos α sin β = 12(sin(α + β) − sin(α − β))cos of alpha sin of beta equals (1) over (2) (sin of alpha plus beta minus sin of alpha minus beta)

Sum to product

• sin α + sin β = 2 sin(α + β2) cos(α − β2)sin of alpha plus sin of beta equals 2 sin of (alpha plus beta) over (2) cos of (alpha minus beta) over (2)
• sin α − sin β = 2 cos(α + β2) sin(α − β2)sin of alpha minus sin of beta equals 2 cos of (alpha plus beta) over (2) sin of (alpha minus beta) over (2)
• cos α + cos β = 2 cos(α + β2) cos(α − β2)cos of alpha plus cos of beta equals 2 cos of (alpha plus beta) over (2) cos of (alpha minus beta) over (2)
• cos α − cos β = −2 sin(α + β2) sin(α − β2)cos of alpha minus cos of beta equals minus 2 sin of (alpha plus beta) over (2) sin of (alpha minus beta) over (2)

Any triangle

Sides a, b and c are opposite the angles A, B and C.

• Law of sinessin Aa = sin Bb = sin Cc(sin of A) over (a) equals (sin of B) over (b) equals (sin of C) over (c)
• Law of cosinesa² = b² + c² − 2bc cos Aa squared equals b squared plus c squared minus 2 b c cos of A
• b² = a² + c² − 2ac cos Bb squared equals a squared plus c squared minus 2 a c cos of B
• c² = a² + b² − 2ab cos Cc squared equals a squared plus b squared minus 2 a b cos of C
• Law of tangentsa − ba + b = tan(A − B2)tan(A + B2)(a minus b) over (a plus b) equals (tan of (A minus B) over (2)) over (tan of (A plus B) over (2))
• AreaArea = 12ab sin CArea equals (1) over (2) a b sin of C