Trig Identities

itsclaire9

New member
Joined
Feb 24, 2012
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I have a take-home test due Monday and this one problem looks deathly to me. We're learning about solving Identities in Trig right now, all that sin and cos stuff. Here I've provided a picture of the problem as it would be really difficult to clearly type it out (I've tried already haha)

http://i726.photobucket.com/albums/ww263/itsclaire9/Photoon2-24-12at1119AM.jpg
http://i726.photobucket.com/albums/ww263/itsclaire9/Photoon2-24-12at1119AM.jpg

View attachment 1761

Please help!

Are you allowed to seek external help for take-home test?
 
Yeah, I know it seems counter-intuitive but my teacher's exact words were: "You can use your friends, you can use your notes, you can use the text book, you can use your tutor, but you can't use me as your teacher." We're allowed to use external sources for help.
 
To get help on this site, you have to share your work and tell us where you are stuck.

Hint to start:

tan-1(x) = Θ → tan(Θ) = x
 
From that hint of tan-1(x) = Θ → tan(Θ) = x, I've expanded it to this

sec { [tan(Θ) = (-4/3)] - [csc(Θ) = (-13/12)] } cot2 (2tan(Θ) = 4/3)

Was that the right direction you were trying to get me to?
My first instincts would tell me to draw out the triangles because they're all pythagorean triples.
 
From that hint of tan-1(x) = Θ → tan(Θ) = x, I've expanded it to this

sec { [tan(Θ) = (-4/3)] - [csc(Θ) = (-13/12)] } cot2 (2tan(Θ) = 4/3)

Was that the right direction you were trying to get me to?
My first instincts would tell me to draw out the triangles because they're all pythagorean triples.

No...

If tan-1(-4/3) = Θ ...→... then ... tan(Θ) = -4/3 → sin(Θ) = - 4/5 and cos(Θ) = 3/5 .... assuming -π/2 < Θ < π/2

If csc-1(-13/12) = Φ ... → ... then ... csc(Φ) = -13/12 → sin(Φ) = - 12/13 → cos(Φ) = 5/13 .... assuming -π/2 < Φ < π/2

Then

sec[tan-1(-4/3) - csc-1(-13/12)] = sec[Θ-Φ] = 1/cos(Θ-Φ) ..... and continue
 
So I pulled it out as:

1/cosΘ (Θ - Φ) =(3/5)(5/13) + (-4/5)(-12/13)

and I ended up getting 63/65 as an answer
 
Nevermind, I got -33/65 so I flipped it to -65/33 because secant is the reciprocal of cosine.
 
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