Expert: Paul Klarreich - 5/8/2007

Question
I find in trouble in solving these trigonometric equations : Can you help me ?

a) sin2x-cosx = 0
b)sin(2x-10°) = 1/2
c) cosx2x-sin^2 ( x/2)+3/4  = 0
d)sin^2t = cos^2 (t)  + 1/2   ( 't' stands for theta ).
e tan ( x+15° ) = 3 tan x


Thank you


Answer
Questioner:   Enrico
Category:  Advanced Math
 
Subject:  Plane Trigonometry
Question:  I find in trouble in solving these trigonometric equations : Can you help me ?

a) sin2x-cosx = 0
b)sin(2x-10°) = 1/2
c) cosx2x-sin^2 ( x/2)+3/4  = 0
d)sin^2t = cos^2 (t)  + 1/2   ( 't' stands for theta ).
e tan ( x+15° ) = 3 tan x


Thank you
.....................................
Hi, Enrico,

This is a lot of questions, so I'll try to get you started on them :

a) sin2x-cosx = 0

Use the identity  sin(2x) = 2 sin x cos x:

2 sin x cos x - cos x = 0

Now factor the left side:

cos x(2 sin x - 1) = 0

and treat it as if it were a quadratic equation, setting each factor equal to zero:

cos x = 0  gives you solutions  x = pi/2, 3pi/2

2 sin x - 1 = 0  gives you  sin x = 1/2, and solutions in quadrants I and II, related to pi/6.
...............................
b)  sin(2x-10°) = 1/2

Set  @ = 2x - 10  and solve:

sin @ = 1/2

That gives you values of  30 and 150 degrees.  Now set  2x - 10 = 30,150 and solve for x.
.....................................
c) cosx2x-sin^2 ( x/2)+3/4  = 0

THIS IS GARBLED.  I don't know what that first term is.
................................
d)sin^2t = cos^2 (t)  + 1/2   ( 't' stands for theta ).

Be careful entering expressions.  I assume the first term is  sin^2(t).

sin^2(t) = cos^2 (t)  + 1/2

-1/2 = cos^2(t) - sin^2(t)

Now use the identity:  

cos(2t) = cos^2(t) - sin^2(t)

cos(2t) = -1/2

That will give you 2t = 2pi/3  and 4pi/3, and you can solve for t.
...............................
e) tan ( x+15° ) = 3 tan x

There are sum-and-difference-of-angles formulas:
             tan A + tan B
tan(A + B) = ---------------
            1 - tan A tan B

             tan A - tan B
tan(A - B) = ---------------
            1 + tan A tan B
 
which you should be able to use here and get exact solutions:

             tan x + tan 15
tan(x + 15) = ---------------
            1 - tan x tan 15

Set that equal to 3 tan x:

          tan x + tan 15
3 tan x = ---------------
         1 - tan x tan 15

Now the rest of this is messy, and I don't have a complete solution for you, but we can get an exact value of tan 15:

**************** GETTING A VALUE FOR  TAN(15) ***********
15 = 60 - 45

tan 45 = 1,  tan 60 = sqrt(3)

             tan A - tan B
tan(A - B) = ---------------
            1 + tan A tan B

               tan 60 - tan 45
tan(60 - 45) = -----------------
              1 + tan 60 tan 45

         sqrt(3) - 1
tan(15) = --------------
         1 + sqrt(3)(1)

         sqrt(3) - 1
tan(15) = -----------
         sqrt(3) + 1

         sqrt(3) - 1 sqrt(3) - 1  
tan(15) = ----------- ------------
         sqrt(3) + 1 sqrt(3) - 1


         3 + 1 - 2 sqrt(3)
tan(15) = -----------------
          3 - 1

         4 - 2 sqrt(3)
tan(15) = -----------------
          2

tan(15) = 2 - sqrt(3)