Expert: Paul Klarreich - 2/7/2008

Question
Solve each of the following equations for 0<t<pi giving all answers exactly in terms of pi. (where t= theta)
i) cos(5t) + sin(5t) + cos(3t) + sin(3t) = 0 and
ii) sin(5t) - sin(2t) =0.


Answer
Questioner:   Jamie
Category:  Advanced Math
Private:  No
 
Subject:  Trigonometry
Question:  Solve each of the following equations for 0<t<pi giving all answers exactly in terms of pi. (where t= theta)
i) cos(5t) + sin(5t) + cos(3t) + sin(3t) = 0 and
ii) sin(5t) - sin(2t) =0.
.............................
Hi, Jamie,

These look like applications of sum-to-product formulas, found at:

www.sosmath.com/trig/Trig5/trig5/trig5.html

Convert these:

cos(5t) + sin(5t) + cos(3t) + sin(3t)
................    .................
cos(5t) + cos(3t) + sin(5t) + sin(3t)
(cos u + cos v)      (sin u + sin v)
................    ................

2 cos(4t) cos (t) + 2 sin(4t) cos t

Factor:

2cos t(cos 4t + sin 4t)

Now set each factor equal to zero and solve:

cos t = 0 --> t = pi/2

cos 4t + sin 4t = 0
sin 4t = - cos 4t

tan 4t = -1  

4t = 3pi/4  or 7pi/4,

t = 3pi/16  or 7pi/16
-----------------------------------
Second:

sin 5t - sin 2t

Do the same, using  sin u - sin v:

2 cos (7t/2) sin (3t/2)
...........
Set cos (7t/2) = 0

7t/2 = pi/2 or  3pi/2

7t = pi or  3pi

t = pi/7 or  3pi/7
.........
Set sin (3t/2) = 0

3t/2 = 0 or pi

3t   = 0 or 2 pi

t = 0 or 2pi/3