Chapter 7 and 8

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Trig Functions

Tangent

Domain: all real numbers except pi/2 + k pi, k is an integer. Range: all real numbers Period = pi x intercepts: x = k pi , where k is an integer. y intercepts: y = 0 symmetry: since tan(-x) = - tan(x) then tan (x) is an odd function and its graph is symmetric with respect the origin. intervals of increase/decrease: over one period and from -pi/2 to pi/2, tan (x) is increasing. Vertical asymptotes: x = pi/2 + k pi, where k is an integer.

Reciprocal identities: tan u=1/cot u; Pythagorean Identities: 1+tan^2u=sec^2u; Quotient Identities: tan u=sin u/cos u; Co-Function Identities: tan(pi/2-u)=cot u

Cotangent

Domain: all real numbers except k pi, k is an integer. Range: all real numbers Period = pi x intercepts: x = pi /2 + k pi , where k is an integer. symmetry: since cot(-x) = - cot(x) then cot (x) is an odd function and its graph is symmetric with respect the origin. intervals of increase/decrease: over one period and from 0 to pi, cot (x) is decreasing. Vertical asymptotes: x = k pi, where k is an integer.

Reciprocal identities: cot u=1/tan u; Pythagorean Identities: 1+cot^2u=csc^2u; Quotient Identities: cot u= cos u/sin u; Co-Function Identities: cot(pi/2-u)=tan u;

Cosecant

Domain: All real number except kpi, k is an interger; Range: (-∞,-1] U [1,∞); Period: 2pi; Symmetry: since csc(-x)=-csc(x) then csc(x) is an odd funtion and its graph is symmetric with respect the origin; intervals of increase/decrease: over one period and from 0 to 2pi, csc (x) is decreasing on (0 , pi/2) U (3pi/2 , 2pi) and increasing on (pi/2 , pi) U (pi / 3pi/2); Vertical asymptotes: x = k pi, where k is an integer.

Reciprocal identities: csc u=1/sin u; Pythagorean Identities: 1+cot^2u=csc^2u; Co-Function Identities: csc(pi/2-u)=sec u;

Sine

Domain: All real numbers; Range: [-1,1]; Period: 2pi; x-intercepts: x=kpi, where k is an integer; y-intercepts: y=0; maximum points: (pi/2 +2kpi, 1), where k is an integer; minimim points: (3pi/2 + 2kpi, -1), where k is an integer; symmetry: since sin(-x)=-sin(x) then sin (x) is an odd function and its graph is symmestric with the respect to the origin (0,0); intervals of increase/decrease: over one period and from 0 to 2pi, sin (x) is increasing on the intervals (0,pi/2) and (3pi/2,2pi), and decreasing on the interval (pi/2, 3pi/2).

Reciprocal identities: sin u=1/csc u; Pythagorean Identities: sin^2u+cos^2=1; Co-Function Identities:sin(pi/2-u)=cos u; Even-Odd Identities: sin(-u)=-sin u; Sum-Difference Formulas: (sin u+/-v) sin u cos v +/- cos u sin v; Double Angle Formulas: sin(2u)=2sin u cos u; Sum-to-Product Formulas: sin u +sin v= 2sin (u+v/2) cos (u-v/2); sin u-sin v=2cos (u+v/2) sin (u-v/2); Product-to-Sum Formulas: sin u sin v= 1/2[cos(u-v)-cos(u+v)]

Secant

Domain: all real numbers except pi/2 + k pi, n is an integer. Range: (-infinity , -1] U [1 , +infinity) Period = 2 pi y intercepts: y = 1 symmetry: since sec(-x) = sec (x) then sec (x) is an even function and its graph is symmetric with respect to the y axis. intervals of increase/decrease: over one period and from 0 to 2 pi, sec (x) is increasing on (0 , pi/2) U (pi/2 , pi) and decreasing on (pi , 3pi/2) U (3pi/2 , 2pi). Vertical asymptotes: x = pi/2 + k pi, where k is an integer.

Reciprocal identities: sec u= 1/cos u; Pythagorean Identities: 1+tan^2u=sec^2u; Co-Function Identities: sec(pi/2-u)=csc u;

Cosine

Domain: all real numbers Range: [-1 , 1] Period = 2pi x intercepts: x = pi/2 + k pi , where k is an integer. y intercepts: y = 1 maximum points: (2 k pi , 1) , where k is an integer. minimum points: (pi + 2 k pi , -1) , where k is an integer. symmetry: since cos(-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to the y axis. intervals of increase/decrease: over one period and from 0 to 2pi, cos (x) is decreasing on (0 , pi) increasing on (pi , 2pi).

Reciprocal identities: cos u= 1/sec u; Pythagorean Identities: sin^2u+cos^2u=1; Quotient Identities: cot u = cos u/sin u; Co-Function Identities: cos(pi/2-u)=sin u

Chapter 7 and 8