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

The x-coordinate of p is called the cosine of A và is denoted by cos A;

The y-coordinate of p. Is called the sine of A và is denoted by sin A;

The number \$fracsin(A)cos(A)\$ is called the tangent of A & is denoted by chảy 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>.

The tangent function

tan : R -> RThe range of the function is R.The period is \$pi\$ and 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\$

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\$ & 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\$
\$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 and 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\$ và \$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\$ and \$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 chảy \$ \$- extrm tung alpha\$ \$ extrm cot alpha\$ \$- extrm cot alpha\$ \$- extrm rã alpha\$ \$ extrm cot \$ \$- extrm cot alpha\$ \$ extrm tan alpha\$ \$- extrm tan 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 và 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 = frac14ig(sin(alpha+eta-gamma)+sin(eta+gamma-alpha)+sin(gamma+alpha-eta)-sin(alpha+eta+gamma)ig)\$

\$cos alphacos etacos gamma = frac14ig(cos(alpha+eta-gamma)+cos(eta+gamma-alpha)+cos(gamma+alpha-eta)+cos(alpha+eta+gamma)ig)\$

\$sin alphasin etacos gamma = frac14ig(-cos(alpha+eta-gamma)+cos(eta+gamma-alpha)+cos(gamma+alpha-eta)-cos(alpha+eta+gamma)ig)\$

\$sin alphacos etacos gamma = frac14ig(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^2ig(fracpi4pm fracalpha2ig)=2cos^2ig(fracpi4mp fracalpha2ig)\$