# Math Help - trig problem: circular functions of real numbers

1. ## trig problem: circular functions of real numbers

Solve the equation for 0 (less than or equal to) x (less than or equal to) 2pi.

1. sin x = sin (x+2)
2. cos x = cos (x+1)

2. Originally Posted by lolo

Solve the equation for 0 (less than or equal to) x (less than or equal to) 2pi.

1. sin x = sin (x+2)
2. cos x = cos (x+1)
sinX = sin(X +2)

Normally, it would be X = X +2.
So do it by expansion:

sinX = sinXcos(2) +cosXsin(2)
sinX -sinXcos(2) = cosXsin(2)
sinX(1 -cos(2)) = cosXsin(2)
Divide both sides by cosX,
tanX(1 -cos(2)) = sin(2)
tanX = sin(2) / (1 -cos(2))
Using the calculator,
tanX = 0.642096 -------------positive tan, so X is in the 1st or 3rd quadrants.

X = arctan(0.642096) = 0.570796 radian

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You should be able to solve the second equation now.

You should find that X is in the 2nd or 4th quadrants,
and X = 2.641593 or 5.783185 radians.

3. Originally Posted by ticbol
sinX = sin(X +2)

Normally, it would be X = X +2.
So do it by expansion:

sinX = sinXcos(2) +cosXsin(2)
sinX -sinXcos(2) = cosXsin(2)
sinX(1 -cos(2)) = cosXsin(2)
Divide both sides by cosX,
tanX(1 -cos(2)) = sin(2)
tanX = sin(2) / (1 -cos(2))
Using the calculator,
tanX = 0.642096 -------------positive tan, so X is in the 1st or 3rd quadrants.

X = arctan(0.642096) = 0.570796 radian

------------------------------------
You should be able to solve the second equation now.

You should find that X is in the 2nd or 4th quadrants,
and X = 2.641593 or 5.783185 radians.
thanks for replying! I found out it's the right answer, but I don't understand where the cos came from when you expanded sin(x+2), can you explain it please?

4. Originally Posted by lolo
thanks for replying! I found out it's the right answer, but I don't understand where the cos came from when you expanded sin(x+2), can you explain it please?
You mean the following?
sin(X +1) = sinXcos1 +cosXsin1 ?

You thought that should have been
sin(X +1) = sinX +sin1 ?

The first one is the correct one. That is how it is done with trig functions.

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Example: Find sin(30deg +60deg).

The correct way:
sin(30deg +60deg)
= sin(30deg)cos(60deg) +cos(30deg)sin(60deg)
= (1/2)(1/2) +(sqrt(3)/2)(sqrt(3)/2)
= 1/4 +3/4
= 1

A wrong way:
sin(30deg +60deg)
= sin(30deg) +sin(60deg)
= 1/2 +sqrt(3)/2
= [1 +sqrt(3)] / 2

Well, of course, sin(30deg +60deg) = sin(90deg) = 1.