Different ways of solving for x in trig. equations

[Attis]

Homework Statement

According to my math book, I solved the following trig equation correctly:
1)
cos3x=0,500=
3x = 60° + n360°
x=20+n120°

2)I also solved this problem correctly:
4sin^2x -3sin^x= 0
sinx(4sinx-3)=0
x=n*360°
or
4sinx=3
sinx=3/4
x= ca. 49°
x=49° + n360° or 131+ n360°.

Now I´m wondering why I can´t solve question nr 1) in the following way:
cosx = 0,500/3
x=80°
x=80° + n360°

i.e. why can´t I solve question nr 1 the way I did question nr 2?
I was also wondering if sin2x is the same as 2sinx? does it make a difference where I place the 2?

Homework Equations

NA

The Attempt at a Solution

see above

 

[Curious3141]

Homework Statement

According to my math book, I solved the following trig equation correctly:
1)
cos3x=0,500=
3x = 60° + n360°
x=20+n120°

2)I also solved this problem correctly:
4sin^2x -3sin^x= 0
sinx(4sinx-3)=0
x=n*360°
or
4sinx=3
sinx=3/4
x= ca. 49°
x=49° + n360° or 131+ n360°.

Now I´m wondering why I can´t solve question nr 1) in the following way:
cosx = 0,500/3
x=80°
x=80° + n360°

Because (in general) $\cos nx \neq n\cos x$. In fact the expressions for different values of $n$ higher than 1 (like $n=2$ and $n=3$) are important trigonometric identities.

i.e. why can´t I solve question nr 1 the way I did question nr 2?

In no. 2, you're just factoring out the $\sin x$. That's completely valid. $\sin^2 x$ is the conventional shorthand for $(\sin x)^2$. This is the square of the sine of angle $x$. This is completely different from $\sin 2x$, which is the sine of the angle $2x$.

I was also wondering if sin2x is the same as 2sinx? does it make a difference where I place the 2?

Huge difference, just like for the cosine.

 

[Attis]

Because (in general) $\cos nx \neq n\cos x$. In fact the expressions for different values of $n$ higher than 1 (like $n=2$ and $n=3$) are important trigonometric identities.



In no. 2, you're just factoring out the $\sin x$. That's completely valid. $\sin^2 x$ is the conventional shorthand for $(\sin x)^2$. This is the square of the sine of angle $x$. This is completely different from $\sin 2x$, which is the sine of the angle $2x$.



Huge difference, just like for the cosine.

Perfect. Now I get it! thanks a lot!

 

[Fredrik]

I was also wondering if sin2x is the same as 2sinx? does it make a difference where I place the 2?

Check out the graphs:

To put a 2 in front of sin x is to double the amplitude. To put a 2 in front of the x in sin x is to double the frequency.

Wolfram Alpha is great for stuff like this.

 

[Attis]

Check out the graphs:

To put a 2 in front of sin x is to double the amplitude. To put a 2 in front of the x in sin x is to double the frequency.

Wolfram Alpha is great for stuff like this.

Ok, thanks for explaining!