# [SOLVED] Trig definitions and identities

• Mar 5th 2010, 01:17 PM
thekrown
[SOLVED] Trig definitions and identities
I have my book open here with the definitions and fundamental identities to sine cosine and tangent.

For example, sine theta is y/r and 1/csc theta

It does not list csc, sec and cot. How can I find these?

What if we have arcsin (inverse of sin), what would the csc be?
• Mar 5th 2010, 01:29 PM
harish21
Quote:

Originally Posted by thekrown
I have my book open here with the definitions and fundamental identities to sine cosine and tangent.

For example, sine theta is y/r and 1/csc theta

It does not list csc, sec and cot. How can I find these?

What if we have arcsin (inverse of sin), what would the csc be?

Whilde defining sin(theta), where did you get y and r from? Are you using a right angled triangle with an angle theta to determine the trigonometric identities? Please state clearly.

similarly, csc(theta) = 1/sin(theta)
sec(theta)= 1/cos(theta)
cot(theta) = 1/tan(theta)

arc sin is the same as csc!
arcsin(theta)= (inverse of sin(theta)) = 1/sin(theta) = csc(theta)
• Mar 5th 2010, 01:49 PM
thekrown
Most of our triangles are 45 or 30/60's. The y/r comes from a different of viewing the adjacent/opposite business which I find overlycomplicated...

I use y/r because I simplify the 45 and 30/60 triangles to radius 1 and always end up with the same values.

This might be why I'm having trouble with calculus trig.

The teachers didn't tell us this, so I must somehow know this but I don't. Thank you for your help.

Is it safe to say that csc, sec, cot are inverses of sin, cos, tangent and vice versa?
• Mar 5th 2010, 02:21 PM
masters
Quote:

Originally Posted by thekrown
I have my book open here with the definitions and fundamental identities to sine cosine and tangent.

For example, sine theta is y/r and 1/csc theta

It does not list csc, sec and cot. How can I find these?

What if we have arcsin (inverse of sin), what would the csc be?

Hi thekrown,

Visualize a circle with center (0, 0) and radius = r. Pick a point on the circle in quadrant 1, for example, and call it P(x, y).

Draw a perpendicular to the x-axis from this point. The degree of rotation is the measure of the angle formed by the radius and the x-axis. Let's call this angle $\theta$.

The side opposite this angle is y units long.
The side adjacent to this angle is x units long.
The radius is r units long.

The six trigonometric functions of this angle are defined this way.

$\sin \theta=\frac{y}{r}$ <====> $\csc \theta=\frac{r}{y}$

$\cos\theta=\frac{x}{r}$ <====> $\sec \theta=\frac{r}{x}$

$\tan\theta=\frac{y}{x}$ <====> $\cot \theta=\frac{x}{y}$

To understand arcsin, sometimes written $\sin^{-1}$,
recall just as $y=\sqrt{x}$ is defined such that $y^2=x$,
$y=\arcsin x$ is defined so that $\sin y = x$

Quote:

Originally Posted by harish21
Whilde defining sin(theta), where did you get y and r from? Are you using a right angled triangle with an angle theta to determine the trigonometric identities? Please state clearly.

similarly, csc(theta) = 1/sin(theta)
sec(theta)= 1/cos(theta)
cot(theta) = 1/tan(theta)

arc sin is the same as csc!
arcsin(theta)= (inverse of sin(theta)) = 1/sin(theta) = csc(theta)

Arcsin is not the reciprocal of sine. It's the inverse of sine. The reciprocal of sine (sin) is cosecant (csc).
• Mar 5th 2010, 02:30 PM
Quote:

Originally Posted by harish21
Whilde defining sin(theta), where did you get y and r from? Are you using a right angled triangle with an angle theta to determine the trigonometric identities? Please state clearly.

similarly, csc(theta) = 1/sin(theta)
sec(theta)= 1/cos(theta)
cot(theta) = 1/tan(theta)

arc sin is the same as csc! no!!
arcsin(theta)= (inverse of sin(theta)) = 1/sin(theta) = csc(theta) no! it's not

$arcsin(x)=sin^{-1}(x)$

returns an angle

$\frac{1}{sin(x)}=csc(x)$

The "inverse function" is not the same as the "inverse" of the function,
if the terms are rigorously adhered to.

Edit:
Sorry masters!!
I didn't see you spotted it