t5 graphs of trig functions and inverse trig functions

Graphs of Trig. FunctionsThe graph of y=sin(x)

Graphs of Trig. FunctionsThe graph of y=cos(x)

Graphs of Trig. FunctionsThe graph of y=cos(x)90o180o0o

Graphs of Trig. FunctionsThe graph of y=cos(x)90o180o270o360o0o

Graphs of Trig. FunctionsThe graph of y=cos(x)The graph of y=sin(x)

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.The graph of a periodic function: Frank Ma2006p

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). The graph of a periodic function: Frank Ma2006p

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). The graph of a periodic function: Frank Ma2006pone period

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function: Frank Ma2006pone period

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function: Frank Ma2006pxx+ppone period

Periodic FunctionsGiven a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that f(x) = f(x+b) for all x.The smallest number p>0 such that f(x) = f(x+p) is called the period of f(x). Over every interval of length p, the graph of a periodic function repeats itself. The graph of a periodic function: Frank Ma2006pxx+ppone periodf(x) = f(x+p) for all x

Periodic Functionssin(x) and cos(x) are periodic with period p=2π.

Periodic Functionssin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.For y=cos(x):

Periodic Functionssin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.For y=sin(x):0

Periodic FunctionsThe basic period for: y=sin(x)

Periodic FunctionsThe basic period for: y=sin(x) y=cos(x)1-1

Periodic FunctionsThe basic period for: y=sin(x) y=cos(x)1-1The Graph of Tangent

Periodic FunctionsThe basic period for: y=sin(x) y=cos(x)1-1The Graph of TangentThe function tan(x) is not defined when cos(x) is 0,i.e. when x = ±π/2, ±3π/2, ±5π/2, ..

Periodic FunctionsThe basic period for: y=sin(x) y=cos(x)1-1The Graph of TangentThe function tan(x) is not defined when cos(x) is 0,i.e. when x = ±π/2, ±3π/2, ±5π/2, .. Frank Ma2006As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1,

Periodic FunctionsThe basic period for: y=sin(x) y=cos(x)1-1The Graph of TangentThe function tan(x) is not defined when cos(x) is 0,i.e. when x = ±π/2, ±3π/2, ±5π/2, .. Frank Ma2006As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1, but the values of cos(x) goes from 1 to 0. So tan(x) goes from 0 to +∞.

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x).

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 tan(x)

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞01/313tan(x)

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞01/313-1/3-1-3-∞tan(x)

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞01/313-1/3-1-3-∞tan(x)0π/2 -π/2

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞01/313-1/3-1-3-∞tan(x)The same pattern repeats itself every πinterval. 0π/2 -π/2

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞01/313-1/3-1-3-∞tan(x)The same pattern repeats itself every πinterval. In other words, y = tan(x) is a periodic function with period πas shown in the graph.0π/2 -π/2

The Graph of TangentSince tan(x) is odd, so as the values of x goes from 0 to -π/2 we get the corresonding negative outputs for tan(x). Specifically,xπ/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 ∞01/313-1/3-1-3-∞tan(x)The same pattern repeats itself every πinterval. In other words, y = tan(x) is a periodic function with period πas shown in the graph.π-π0π/2 -π/2 3π/2 -3π/2 y = tan(x)

The Graph of Inverse Trig-FunctionsRecalll that for y = cos-1(x), then 0 < y < π.

The Graph of Inverse Trig-FunctionsRecalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.

The Graph of Inverse Trig-FunctionsRecalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.(-1, π)(0, π/2)(1, 0)-11The graph of y = cos-1(x)

The Graph of Inverse Trig-FunctionsRecalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.(-1, π)(0, π/2)(1, 0)-11The graph of y = cos-1(x)Remark: The above graphs of y = sin-1(x) and y = cos-1(x) are the complete graphs (i.e. that's all there is).

The Graph of Inverse Trig-FunctionsThe domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.

The Graph of Inverse Trig-FunctionsThe domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.x∞01/313-1/3-1-3-∞tan-1(x)

The Graph of Inverse Trig-FunctionsThe domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.x∞01/313-1/3-1-3-∞tan-1(x)π/6 0π/4 π/3 π/2

The Graph of Inverse Trig-FunctionsThe domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.x∞01/313-1/3-1-3-∞tan-1(x)π/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2

The Graph of Inverse Trig-FunctionsThe domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.x∞01/313-1/3-1-3-∞tan-1(x)π/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 Here is the graph of y = tan-1(x)y = π/2 (1,π/4)(0,0)(-1,-π/4)y = -π/2

The Graph of Inverse Trig-FunctionsThe domain of y = tan-1(x) is all real numbers and the output y is restricted to -π/2 < y < π.x∞01/313-1/3-1-3-∞tan-1(x)π/6 0π/4 π/3 -π/2 -π/6 -π/4 -π/3 π/2 Here is the graph of y = tan-1(x)y = π/2 (1,π/4)(0,0)(-1,-π/4)y = -π/2 Remark: y =tan-1(x) has two horizontal asymptoes.

t5 graphs of trig functions and inverse trig functions

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t5 graphs of trig functions and inverse trig functions