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13.1: Basic Functions and Identities

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    Graphs of the Parent Functions

    Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.
    Figure A1
    Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.
    Figure A2
    Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.
    Figure A3

    Graphs of the Trigonometric Functions

    Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.
    Figure A4
    Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.
    Figure A5
    Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.
    Figure A6
    Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.
    Figure A7

    Trigonometric Identities

    Pythagorean Identities cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ
    Even-Odd Identities cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=sinθ tan(−θ)=tanθ csc(−θ)=cscθ cot(−θ)=cotθ cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=sinθ tan(−θ)=tanθ csc(−θ)=cscθ cot(−θ)=cotθ
    Cofunction Identities cosθ=sin( π 2 θ ) sinθ=cos( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ ) cosθ=sin( π 2 θ ) sinθ=cos( π 2 θ ) tanθ=cot( π 2 θ ) cotθ=tan( π 2 θ ) secθ=csc( π 2 θ ) cscθ=sec( π 2 θ )
    Fundamental Identities tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ
    Sum and Difference Identities cos(α+β)=cosαcosβsinαsinβ cos(αβ)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(αβ)=sinαcosβcosαsinβ tan(α+β)= tanα+tanβ 1tanαtanβ tan(αβ)= tanαtanβ 1+tanαtanβ cos(α+β)=cosαcosβsinαsinβ cos(αβ)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(αβ)=sinαcosβcosαsinβ tan(α+β)= tanα+tanβ 1tanαtanβ tan(αβ)= tanαtanβ 1+tanαtanβ
    Double-Angle Formulas sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ sin 2 θ cos(2θ)=12 sin 2 θ cos(2θ)=2 cos 2 θ1 tan(2θ)= 2tanθ 1 tan 2 θ sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ sin 2 θ cos(2θ)=12 sin 2 θ cos(2θ)=2 cos 2 θ1 tan(2θ)= 2tanθ 1 tan 2 θ
    Half-Angle Formulas sin α 2 =± 1cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1cosα sinα sin α 2 =± 1cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1cosα sinα
    Reduction Formulas sin 2 θ= 1cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1cos( 2θ ) 1+cos( 2θ ) sin 2 θ= 1cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1cos( 2θ ) 1+cos( 2θ )
    Product-to-Sum Formulas cosαcosβ= 1 2 [ cos(αβ)+cos(α+β) ] sinαcosβ= 1 2 [ sin(α+β)+sin(αβ) ] sinαsinβ= 1 2 [ cos(αβ)cos(α+β) ] cosαsinβ= 1 2 [ sin(α+β)sin(αβ) ] cosαcosβ= 1 2 [ cos(αβ)+cos(α+β) ] sinαcosβ= 1 2 [ sin(α+β)+sin(αβ) ] sinαsinβ= 1 2 [ cos(αβ)cos(α+β) ] cosαsinβ= 1 2 [ sin(α+β)sin(αβ) ]
    Sum-to-Product Formulas sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 ) sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 )
    Law of Sines sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ
    Law of Cosines a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ
    Table A1

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