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### 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/
```
Associated Topics:
High School Trigonometry

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