Expert: Paul Klarreich - 2/13/2008

Question
Simplify :  

a) ( sin 80°- sin 10° )/(sin 80° + sin 10° )

b)  ( sin 130° + sin 20° )/( cos 130° + cos 20° )

c) Find cos (x+y) when both x and y are acute angles    sin x = 3/5    and  sin y = 5/13. Work out this problem
without using tables.

Answer
Questioner:   Mario
Category:  Advanced Math
Private:  No
 
Subject:  Plane Trigonometry
Question:  Simplify :  

a) ( sin 80°- sin 10° )/(sin 80° + sin 10° )

b)  ( sin 130° + sin 20° )/( cos 130° + cos 20° )

c) Find cos (x+y) when both x and y are acute angles    sin x = 3/5    and  sin y = 5/13. Work out this problem
without using tables.

......................................
Hi, Mario,


a) ( sin 80°- sin 10° )/(sin 80° + sin 10° )

(sin 80°- sin 10° )
----------------------
(sin 80° + sin 10° )

Note:  sin A = cos (90 - A),  so sin 80 = cos 10.

(cos 10°- sin 10° )
--------------------
(cos 10° + sin 10°)

Rationalize:

(cos 10° - sin 10°)(cos 10° - sin 10°)
--------------------------------------
(cos 10° + sin 10°)(cos 10° - sin 10°)

(cos^2(10°) + sin^2(10°) - 2 cos 10°sin 10°
--------------------------------------
    (cos^2(10°) - sin^2(10°))


1 - sin 20°
-----------
  cos 20°
................................
b)( sin 130° + sin 20° )/( cos 130° + cos 20° )

(sin 130° + sin 20°)
--------------------
(cos 130° + cos 20°)
I am not sure what you want here, so how is this:

sin 150°cos 20 - cos 150 sin 20 + sin 20°
-------------------------------------------
cos 150°cos 20 - sin 150 sin 20 + cos 20°

Now  sin 150 = 1/2
cos 150 = - sqrt(3)/2

1/2 cos 20 - sqrt(3)/2 sin 20 + sin 20°
-------------------------------------------
- sqrt(3)/2 cos 20 - 1/2 sin 20 + cos 20°

cos 20 - sqrt(3) sin 20 + 2 sin 20°
-------------------------------------------
- sqrt(3) cos 20 - sin 20 + 2 cos 20°


cos 20  + sin 20(- sqrt(3) + 2)
----------------------------------
cos 20( - sqrt(3) + 2) - sin 20
......................................
c) Find cos (x+y) when both x and y are acute angles    sin x = 3/5    and  sin y = 5/13. Work out this problem without using tables.

This is a standard thing.  For x, make this triangle:

    /|
   / |
5 /  |3
 /   |
/    |
/x    |
------+
  4

Now you can 'read off' cos x = 4/5 from that diagram.
For y, draw this;

    /|
   / |
13 /  |5
 /   |
/    |
/x    |
------+
  12

and 'read off' cos y = 12/13

Now cos(x + y) = cos x cos y - sin x sin y, and put in your stuff.