A circle centered at the origin of the coordinate system và with a radius of 1 is known as a unit circle.

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If p is a point from the circle & A is the angle between PO and x axis then:

The x-coordinate of phường is called the cosine of A and is denoted by cos A;

The y-coordinate of phường is called the sine of A and is denoted by sin A;

The number $fracsin(A)cos(A)$ is called the tangent of A và is denoted by tung A;

The number $fraccos(A)sin(A)$ is called the cotangent of A và is denoted by cot A.


The sine function

sin: R -> RAll trigonometric functions are periodic. The period of sine is $2pi$.The range of the function is <-1,1>.

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The tangent function

tan : R -> RThe range of the function is R.The period is $pi$ & the tangent function is undefined at $x = fracpi2 + kpi$, k=0,1,2,...Here is the graph of the tangent function on the interval $0 - pi$

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Animated graph (opens in a new window):The graph of tangent function on the interval $0 - 2pi$

The cotangent function

cot : R -> RThe range of the function is R.The period is $pi$ and that the function is undefined at $x = kpi$, k=0,1,2,...

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The values of sin, cos, tan, cot at the angles of 0°, 30°, 60°, 90°, 120°,135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°
$A^circ$$0^circ$$30^circ$$45^circ$$60^circ$$90^circ$$120^circ$$135^circ$$150^circ$$180^circ$$210^circ$$225^circ$$240^circ$$270^circ$$300^circ$$315^circ$$330^circ$$360^circ$
$A rad$$0$$fracpi6$$fracpi4$$fracpi3$$fracpi2$$frac2pi3$$frac3pi4$$frac5pi6$$pi$$frac7pi6$$frac5pi4$$frac4pi3$$frac3pi2$$frac5pi3$$frac7pi4$$frac11pi6$$2pi$
$sin A$$0$$frac12$$fracsqrt22$$fracsqrt32$$1$$fracsqrt32$$fracsqrt22$$frac12$$0$$-frac12$$-fracsqrt22$$-fracsqrt32$$-1$$-fracsqrt32$$-fracsqrt22$$-frac12$$0$
$cos A$$1$$fracsqrt32$$fracsqrt22$$frac12$$0$$-frac12$$-fracsqrt22$$-fracsqrt32$$-1$$-fracsqrt32$$-fracsqrt22$$-frac12$$0$$frac12$$fracsqrt22$$fracsqrt32$$1$
$ an A$$0$$fracsqrt33$$1$$sqrt3$$-$$-sqrt3$$-1$$-fracsqrt33$$0$$fracsqrt33$$1$$sqrt3$$-$$-sqrt3$$-1$$-fracsqrt33$$0$$cot A$$-$$sqrt3$$1$$fracsqrt33$$0$$-fracsqrt33$$-1$$-sqrt3$$-$$sqrt3$$1$$fracsqrt33$$0$$-fracsqrt33$$-1$$-sqrt3$$-$

The easiest way khổng lồ remember the basic values of sin và cosat the angles of 0°, 30°, 60°, 90°:sin(<0, 30, 45, 60, 90>) = cos(<90, 60, 45, 30, 0>) = sqrt(<0, 1, 2, 3, 4>/4)

Basic Trigonometric Identities

For every angle $alpha$ corresponds exactly one point $P(cos(alpha),sin(alpha))$ on the unit circle.


$sin^2(alpha) + cos^2(alpha) = 1$

If the sum of two angles $alpha$ & $eta$ is 180 (i.e. $alpha + eta = 180^circ$) then:
$sin(alpha) = sin(eta)$$cos(alpha) = -cos(eta)$$ an(alpha) = - an(eta)$$cot(alpha) = -cot(eta)$

If the sum of two angles $alpha$ & $eta$ is 90 (i.e. $alpha + eta = 90^circ$) then:
$sin(alpha) = cos(eta)$$cos(alpha) = sin(eta)$$ an(alpha) = cot(eta)$$cot(alpha) = an(eta)$
$-alpha$$90^circ - alpha$$90^circ + alpha$$180^circ - alpha$
$ extrm sin $$- extrm sin alpha$$ extrm cos alpha$$ extrm cos alpha$$ extrm sin alpha$
$ extrm cos $$ extrm cos alpha$$ extrm sin alpha$$- extrm sinalpha$$- extrm cos alpha$
$ extrm tan $$- extrm rã alpha$$ extrm cot alpha$$- extrm cot alpha$$- extrm chảy alpha$
$ extrm cot $$- extrm cot alpha$$ extrm chảy alpha$$- extrm rã alpha$$- extrm cot alpha$

Trigonometric Formulas

Half-Angle Formulas

$sinfracalpha2=pmsqrtfrac1-cos alpha2$

+ (positive) if $fracalpha2$ lies in quadrant | or ||

- (negative) if $fracalpha2$ lies in quadrant ||| or |V

$cosfracalpha2=pmsqrtfrac1+cos alpha2$

+ (positive) if $fracalpha2$ lies in quadrant | or |V

- (negative) if $fracalpha2$ lies in quadrant || or |||

$ anfracalpha2=pmsqrtfrac1-cos alpha1+cos alpha$

+ (positive) if $fracalpha2$ lies in quadrant | or |||

- (negative) if $fracalpha2$ lies in quadrant || or |V

$cotfracalpha2=pmsqrtfrac1+cos alpha1-cos alpha$

+ (positive) if $fracalpha2$ lies in quadrant | or |||

- (negative) if $fracalpha2$ lies in quadrant || or |V

$ anfracalpha2 = fracsin alpha1+cos alpha = frac1-cos alphasin alpha=csc alpha-cot alpha$

$cotfracalpha2 = fracsin alpha1-cos alpha = frac1+cos alphasin alpha=csc alpha+cot alpha$


Double & Triple Angle Formulas

$sin(2alpha) = 2sin(alpha)cdot cos(alpha)$

$cos(2alpha) = cos^2(alpha) - sin^2(alpha) = 2cos^2(alpha) - 1 = 1 - 2sin^2(alpha)$

$ an(2alpha) = frac2 an(alpha)1- an^2(alpha)$

$cos(2alpha) = frac1 - an^2(alpha)1 + an^2(alpha)$

$sin(2alpha) = frac2 an(alpha)1 + an^2(alpha)$

$sin3alpha = 3sin alpha - 4 sin^3alpha$

$cos3alpha = 4cos^3alpha - 3 cos alpha$

$ an3alpha=frac3 an alpha - an^3alpha1-3 an^2alpha$

$cot3alpha=fraccot^3alpha-3cot alpha3cot^2alpha-1$

$sin4alpha = 4cos^3Acdot sin alpha - 4cos alphacdot sin^3alpha$

$cos4alpha = cos^4alpha - 6cos^2alphacdot sin^2alpha + sin^4alpha$

$ an4alpha=frac4 an alpha - 4 an^3A1-6 an^2alpha+ an^4alpha$

$cot4alpha=fraccot^4alpha-6cot^2alpha+14cot^3alpha-4cot alpha$

Power-Reducing Formulas

$sin^2(alpha)=frac1 - cos(2alpha)2$

$sin^3(alpha)=frac3sin alpha - sin(3alpha)4$

$sin^4(alpha)=fraccos(4alpha) - 4cos(2alpha) + 38$

$cos^2(alpha) = frac1 + cos(2alpha)2$

$cos^3(alpha)=frac3cos alpha + cos(3alpha)4$

$cos^4(alpha)=frac4cos(2alpha) + cos(4alpha) + 38$

Sum & Difference of Angles

$sin(alpha + eta) = sin(alpha)cdot cos(eta) + cos(alpha)cdot sin(eta)$

$sin(alpha - eta) = sin(alpha)cdot cos(eta) - cos(alpha)cdot sin(eta)$

$cos(alpha + eta) = cos(alpha)cdot cos(eta) - sin(alpha)cdot sin(eta)$

$cos(alpha - eta) = cos(alpha)cdot cos(eta) + sin(alpha)cdot sin(eta)$

$ an(alpha + eta) = fracsin(alpha + eta)cos(alpha + eta)=fracsin(alpha)cdot cos(eta) + cos(alpha)cdot sin(eta)cos(alpha)cdot cos(eta) - sin(alpha)cdot sin(eta)$

$ an(alpha + eta) = frac an(alpha) + an(eta)1 - an(alpha)cdot an(eta)$

$cot(alpha pm eta) = fraccot(eta)cot(alpha)mp 1cot(eta)pm cot(alpha)=frac1mp an(alpha) an(eta) an(alpha)pm an(eta)$

$sin(alpha + eta + gamma) = sin alphacdotcos etacdotcos gamma + cos alphacdotsin etacdotcos gamma + cos alphacdotcos etacdotsin gamma - sin alphacdotsin etacdotsin gamma$

$cos(alpha + eta + gamma) = cos alphacdotcos etacdotcos gamma - sin alphacdotsin etacdotcos gamma - sin alphacdotcos etacdotsin gamma $$- sin alphacdotcos eta cdotsin gamma - cos alpha cdot sin etacdot sin gamma$

$ an(alpha + eta + gamma) = frac an alpha + an eta + an gamma - an alphacdot an eta cdot an gamma1 - an alpha cdot an eta - an etacdot an gamma - an alphacdot an gamma$

Sum & Difference of Trigonometric Functions

$ extrm sin alpha + extrm sin eta = 2 extrm sin fracalpha + eta2 extrm cos fracalpha - eta2$

$ extrm sin alpha - extrm sin eta = 2 extrm sin fracalpha - eta2 extrm cos fracalpha + eta2$

$ extrm cos alpha + extrm cos eta = 2 extrm cos fracalpha + eta2 extrm cos fracalpha - eta2$

$ extrm cos alpha - extrm cos eta = -2 extrm sin fracalpha + eta2 extrm sin fracalpha - eta2$

$ an alpha + an eta = fracsin(alpha+eta)cos alpha cdotcos eta$

$ an alpha - an eta = fracsin(alpha-eta)cos alphacdotcos eta$

$cot alpha + cot eta = fracsin(alpha+eta)sin alphacdotsin eta$

$cot alpha - cot eta = frac-sin(alpha-eta)sin alphacdotsin eta$

Multiplication of 2 Trigonometric Functions

$ extrm sin alpha extrm sin eta = frac12 ( extrm cos (alpha - eta) - extrm cos (alpha + eta))$

$ extrm cos alpha extrm cos eta = frac12 ( extrm cos (alpha - eta) + extrm cos (alpha + eta))$

$ extrm sin alpha extrm cos eta = frac12 ( extrm sin (alpha + eta) + extrm sin (alpha - eta))$

$ an alpha cdot an eta = frac an alpha+ an etacot alpha+cot eta=-frac an alpha- an etacot alpha-cot eta$

$cot alpha cdot cot eta = fraccot alpha+cot eta an alpha+ an eta$

$ an alpha cdot cot eta = frac an alpha+cot etacot alpha+ an eta$

$sin alphasin etasin gamma = frac14ig(sin(alpha+eta-gamma)+sin(eta+gamma-alpha)+sin(gamma+alpha-eta)-sin(alpha+eta+gamma)ig)$

$cos alphacos etacos gamma = frac14ig(cos(alpha+eta-gamma)+cos(eta+gamma-alpha)+cos(gamma+alpha-eta)+cos(alpha+eta+gamma)ig)$

$sin alphasin etacos gamma = frac14ig(-cos(alpha+eta-gamma)+cos(eta+gamma-alpha)+cos(gamma+alpha-eta)-cos(alpha+eta+gamma)ig)$

$sin alphacos etacos gamma = frac14ig(sin(alpha+eta-gamma)-sin(eta+gamma-alpha)+sin(gamma+alpha-eta)+sin(alpha+eta+gamma)ig)$

Tangent half-angle substitution

$sin alpha = frac2 anfracalpha21+ an^2fracalpha2$

$cos alpha = frac1- an^2fracalpha21+ an^2fracalpha2$

$ an alpha = frac2 anfracalpha21- an^2fracalpha2$

$cot alpha = frac1- an^2fracalpha22 anfracalpha2$

Other Trigonometric Formulas

$1pmsin alpha=2sin^2ig(fracpi4pm fracalpha2ig)=2cos^2ig(fracpi4mp fracalpha2ig)$