Calculate sin75 without calculator

[Robin04]

Homework Statement

Calculate sin75 without calculator![/B]

Homework Equations

The Attempt at a Solution

I constructed a triangle with angles 15, 75 and 90 then I defined sin75 by the sides. In a previous exercise the task was the same but with tg15 and I used the result in this execise too.
I wrote the equation: tg15 = b/a and if I express b and put it in the Pythagorean theorem I get the right result. However, if I express a and put it in the Pythagorean theorem I got a wrong result and I don't know why is this happening. I looked through it and I haven't find any calculation error... The second way is a lot longer but I should work. :(

http://kepfeltoltes.hu/thumb/150902/IMG_0122_www.kepfeltoltes.hu_.jpghttp://kepfeltoltes.hu/thumb/150902/IMG_0123_www.kepfeltoltes.hu_.jpg

 

[Chestermiller]

Are you familiar with double angle and half angle formulas? How is sin75 related to cos15? What is cos(2θ) in terms of cosθ?

Chet

 

[Ray Vickson]

I know that this is not a physics homework, but I didn't find any subforum for my trigonometry math homework.

1. Homework Statement
Calculate sin75 without calculator!

Homework Equations

The Attempt at a Solution

I constructed a triangle with angles 15, 75 and 90 then I defined sin75 by the sides. In a previous exercise the task was the same but with tg15 and I used the result in this execise too.
I wrote the equation: tg15 = b/a and if I express b and put it in the Pythagorean theorem I get the right result. However, if I express a and put it in the Pythagorean theorem I got a wrong result and I don't know why is this happening. I looked through it and I haven't find any calculation error... The second way is a lot longer but I should work. :(

http://kepfeltoltes.hu/thumb/150902/IMG_0122_www.kepfeltoltes.hu_.jpghttp://kepfeltoltes.hu/thumb/150902/IMG_0123_www.kepfeltoltes.hu_.jpg

You really should type out your work; most helpers here (me included) will ignore any of the details in your post.

I am willing to offer a hint, though: if you look on-line you can find documents that give closed-form formulas for sines or cosines of all angles from 1 degree to 89 degrees in increments of 1 degree.

 

[Robin04]

Are you familiar with double angle and half angle formulas? How is sin75 related to cos15? What is cos(2θ) in terms of cosθ?

Chet

I haven't learned them but I have the formulas in my book.

 

[Hamza Abbasi]

Break the angle 75 into component angles which are more familiar e.g 15,30 ,45 , then apply sin(alpha+beta) formula...
sin 75
= sin(45 + 30)
= sin45cos30 + cos45sin30
= (½√2)(½√3) + (½√2)(½)
= ¼(√6 + √2)

 

[Ray Vickson]

Break the angle 75 into component angles which are more familiar e.g 15,30 ,45 , then apply sin(alpha+beta) formula...
sin 75
= sin(45 + 30)
= sin45cos30 + cos45sin30
= (½√2)(½√3) + (½√2)(½)
= ¼(√6 + √2)

PF helpers are not allowed to give complete solutions!

 

[Hamza Abbasi]

PF helpers are not allowed to give complete solutions!

Oh ! Sorry I didn't knew that , I will try to avoid it in the future.

 

[Robin04]

Thanks for your answer! :) But can you help me with explaining why my solution doesn't work? They seem completely equivalent for me.