sec, cosec, tan!?

Question

how do i express the following functions in terms of sinx alone??

a. sec(2x)

b. cosec(2x) / tanx

c. tan(2x) assuming π < x < 3π/2



i think a.= 1 / sin(π/2 - 2x). but is that in terms of sinx alone??

thanks

Answer 1

a)
sec x = 1/ cos 2x = 1/(1- 2 sin^2 x)

b) cosec 2x / tan x = (1/ sin 2x) / tan x

= 1/(2 sin x cos x tan x)
= 1(2 sin ^2 x)

c) tan 2x = sin 2x /(cos 2x) = 2 sin x cos x/(1- 2 sin ^2 x)

= 2 sin x sqrt(1- sin ^ 2x) / (1- 2 sin ^ 2 x) x <=5pi/4
= - 2 sin x sqrt(1- sin ^ 2x) / (1- 2 sin ^ 2 x) x >= 5pi/4

becuase of square root it is + or -
becuase x between pi and 3pi/2 so 2x between 2pi and 3pi

if x < 5pi/4 then positive else

-ve

Answer 2

well, some basic trig identities that you need to know to understand those questions are:
csc=1/sin
sec=1/cos
cot=1/tan
tan=sin/cos

also, i think the teacher wants it in terms of sine AND cosine, since you can't simplify to just sines.

try to do it yourself by substituting what I have told you above to get it in terms of sine and cosine








a. substitute sec for 1/cos to get 1/cos(2x)
b. substitute to get 1/sin(2x)/sin(x)/cos(x)
c. substitute to get sin(2x)/cos(2x)

not sure where you get "a.=1 / sin(π/2 - 2x)", as 2x is the angle, you dont need to do anything to it, just leave it as 2x.

remember you are just changing the way it looks, not the actual equation, graph question a and answer a and they should be the same graph, the same goes for b and c. If they aren't, then you did something wrong, or you didn't use parenthesis right in the graphing calculator.

Answer 3

1/tan = cot
1/cos = cosec
1/sin = sec
tan = sin/cos (I always get them mixed up.)

So,

a) 1/sin(2x)
b) (1/cos(2x))/(sin(x)/cos(x))
c) sin(2x)/cos(2x)

Now, replace all the cos x with sqrt(1-Sin(x)^2), and you're in business.

Answer 4

a. sec(2x) = 1/cos(2x) = 1/(1-2sin^2 2x)

b. cosec(2x) / tanx = 1/((sin2x)*sinx/cosx)
= 1/ (2sinx cos x)*sinx/cosx= 1/(2sin^2 x)

c. tan(2x) assuming π < x < 3π/2
= 2tan x/(1-tan^2 x) = (2 sinx/cosx)/( 1 -sin^2 x/cos^2 x)
= (2sinxcos x)/ cos^2x *cos^2x/cos^2x - sin^2x
= sin(2x)/(cos^2x-sin^2 x)
= sin(2x)/(1-2sin^2x)