# Thread: [SOLVED] Trig definitions and identities

1. ## [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?

2. 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)

3. 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?

4. 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 $\displaystyle \theta$.

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.

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

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

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

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

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).

5. 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
$\displaystyle arcsin(x)=sin^{-1}(x)$

returns an angle

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

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