True of false a the period of the cosine function y cos the

True of false:

a. the period of the cosine function, y = cos theta, is 2pi.

b. the period of the tangent function, y = tan theta, is 2pi.

c. the domain of the tangent function, y = tan theta, is all real numbers.

d. the range of the sine function, y = sin theta, is all real numbers.

e. the graph of the function, y = cos theta has no asymptotes.

f. the function y = sec theta is undefined for theta = pi/2     Please explain answers.

Solution

a) the period of the cosine function, y = cos theta

f(cos theta+2*pi)=cos theta ( since if f(x+p)=f(x) ther period is p)

where the period of the cosine function, y = cos theta is 2*pi

b) the period of the tangent function, y = tan theta,

(tan theta+pi)=tan theta

therefore the period of the tangent function, is pi

c)the domain of the tangent function y=tan theta is all real numbers except the values where cos theta is equal to 0.

that is the values pi/2+n*pi for all integers n.

d) the range of the sine function, y = sin theta has a minimum value of -1, and maximum of +1

-1 lessthan or equal to sine theat  lessthan or equal to +1 range is [-1,+1]

e) the graph of the function, y = cos theta

Remember that an asymptote is a line that the graph of a function approaches but never touches.Rational functions contain asymtotes.so y=cos theta has no asymptotes.

f) the function y = sec theta

y=1/(cos theta)

y=1/(cos(pi/2))=1/0=not defining since cos(pi/2)=0