# Pythagorean Identities - finding sin and cos ?

## Homework Statement

For tan(theta) = /2/3 (theta in quadrant 2) find sin(pheta) and cos(pheta) using the pythagorean identities.

## The Attempt at a Solution

Which 2 pythagorean identities would I use?

My trouble is with finding sin and cos from tan.

Dick
Homework Helper
You probably want to use tan(theta)=sin(theta)/cos(theta) and sin(theta)^2+cos(theta)^2=1. Now try it.

You probably want to use tan(theta)=sin(theta)/cos(theta) and sin(theta)^2+cos(theta)^2=1. Now try it.

So, I get this one here: sin(theta)^2+cos(theta)^2=1 - worked out :)

But my other 2 options, according to my text's formula sheet, I can choose from the following formulas?

1+tan^2(theta) = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

?

Dick
Homework Helper
So, I get this one here: sin(theta)^2+cos(theta)^2=1 - worked out :)

But my other 2 options, according to my text's formula sheet, I can choose from the following formulas?

1+tan^2(theta) = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

?

Since your problem involves tan(theta), I'd try working with the first one. sec(theta)=1/cos(theta), yes?

Dick, there is something I am not getting.

My formula sheet has the following 2 formulas to use, and the formula you are telling me is not one of them?

1+tan^2(theta) = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

Thanks so much for your help.

Dick
Homework Helper
Dick, there is something I am not getting.

My formula sheet has the following 2 formulas to use, and the formula you are telling me is not one of them?

1+tan^2(theta) = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

Thanks so much for your help.

sec(theta)=1/cos(theta) may not be in your list of pythogorean identities along with tan(theta)=sin(theta)/cos(theta), but I think you can use them anyway. They are sort of the definition of sec(theta) and tan(theta).

eumyang
Homework Helper
I guess you could start by using the
1 + tan2 θ = sec2 θ
identity to find sec θ, and by extension, cos θ. Then use another Pythagorean identity to find sin θ.

SteamKing
Staff Emeritus
Homework Helper
Take the Pythagorean Identity. Divide thru using the sin^2 term. Simplify.
Take the Pythagorean Identity. Divide thru using the cos^2 term. Simplify.
You should derive the two formulas on your formula sheet. They are both restatements of this most important trig identity.

HallsofIvy
Homework Helper
The 'basic' Pythagorean identity is just the Pythagorean theorem- if a right triangle has legs of length a and b and hypotenuse of length c, then $c^2= a^2+ b^2$.

If tan(theta)= 2/3, then the triangle is similar to a triangle with "opposite side" of length 2 and "near side" of length 3. By the Pythagorean theorem, the hypotenuse, c, is given by $c^2= 2^2+ 3^2= 4+ 9= 13$ so the length of the hypotenuse is $\sqrt{13}$. Now, knowing that you have a right triangle with "opposite side" of length 2, "near side" of length 3, and "hypotenuse" of length $\sqrt{13}$ you can immediately calculate any of the trigonometric functions of this angle.

tiny-tim
Homework Helper
hi nukeman!

(have a theta: θ and try using the X2 icon just above the Reply box )
So, I get this one here: sin(theta)^2+cos(theta)^2=1 - worked out :)

But my other 2 options, according to my text's formula sheet, I can choose from the following formulas?

1+tan^2(theta) = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

?

each of the three Pythagorean identities only relates two functions at a time …

one relates tan and sec, one relates cot and cosec, and one relates cos and sin …

to start from tan and get cos, you need to go from tan to sec (as eumyang says), then use sec = 1/cos

to start from tan and get sin, you either need to do al that, and then go from cos to sin, or you can start again, use cot = 1/tan, then go from cot to cosec, then use sin = 1/cosec