Question

Using the pythagorean identity, prove (sin 0) / sqrt(1-sin^2 0) = tan 0

In which quadrants is this equation true ?

Answer 1

@phi

Answer 2

@jigglypuff314

Answer 3

I assume you mean \( \theta\) and not 0, right ? just use x for the angle.

Answer 4

yes.

Answer 5

\[ \frac{\sin x }{\sqrt{1- \cos^2 x} }= \tan x \]
?

Answer 6

sin*

Answer 7

\[ \frac{\sin x }{\sqrt{1- \sin^2 x} }= \tan x \]

by pythagorean identify, I assume they mean use
\[ \sin^2 x + \cos^2 x= 1 \]
to simplify the bottom square root stuff.

Answer 8

yes.

Answer 9

can you simplify the bottom part ?

Answer 10

idk how to

Answer 11

in
sin^2 x + cos^2 x = 1
add -sin^2 x to both sides and simplify

\[ \sin^2 x - \sin^2 x+ \cos^2 x = 1-\sin^2 x \\\cos^2 x = 1-\sin^2 x\]

Answer 12

ok and then

Answer 13

do you see that you can replace 1- sin^2 x with cos^2 x ?

Answer 14

ohh yes

Answer 15

what do you get if you "swap out" 1-sin^2 x with cos^2 x ?

Answer 16

(sin theta) / sqrt (cos^2 theta) = tan theta

Answer 17

ok, and what is the square root of a square ?

Answer 18

im so lost. what do you mean ?

Answer 19

if you had
\[ \sqrt{x^2} = x\]
we use that same idea for sqr(cos^2 x)

Answer 20

oh ok.

Answer 21

in other words
\[ \sqrt{\cos^2 x} = \cos x \]

Answer 22

ok

Answer 23

what do you have now ?

Answer 24

sin theta / cos theta = tan theta

Answer 25

yes. but (unfortunately) there is a small detail. we assume we take the positive value of the square root. we should write the bottom as | cos x | (absolute value ) to show this

on the right side, tan x = sin x / cos x (by definition)
so you have
\[ \frac{\sin x}{ | \cos x | } = \frac{\sin x }{\cos x} \]
those will be identical as long as cos x is positive (when cos x is negative, the left side with the absolute values will make it positive, and different from the right side)

Answer 26

so that identity is true when cos is positive. any idea what quadrants where cos is positive ?

Answer 27

ohhh

Answer 28

I remember that "cos" goes with "x"