Co-functions and Solving Trig Equations

Date: 11/18/2004 at 01:37:45
From: sara
Subject: Trigonometry

If sin(3x - 26)° = cos(5x - 60)°, find x.

I tried:

  3x - 26 = 5x - 60
       2x = 60 - 26
       2x = 34
        x = 17

but it's not the right answer.

Date: 11/18/2004 at 08:56:32
From: Doctor Ian
Subject: Re: Trigonometry

Hi Sara,

Note that you can't just set the two arguments equal to each other. 
That would only work if 

  sin(A) = cos(A)

Let's look at a right triangle, like this one:

              .
            . .
       1  . B .
        .     . u
      .       .
    . A       .
  . . . . . . .

We know that 

  sin(A) = opposite/hypotenuse, so sin(A) = u/1 = u
  cos(B) = adjacent/hypotenuse, so cos(B) = u/1 = u

So sin(A) = cos(B) when A and B are complementary angles (two angles
that add to make 90 degrees).  This means that if

  sin(3x - 26) = cos(5x - 60)
  
then

  (3x - 26) + (5x - 60) = 90

since the two angles must be complementary.

Does this make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/