1. ## Solving Trig Identities??

For my homework tonight, there is this one problem that is supposed to be easy but i can't figure it out.

Solve for exact solutions in the interval [0degrees, 360degrees)
[ means include, & ) means exclude
2sinx -1= cscx
I am supposed to find the degree value for x. and i think they want me to use cscx=1/sinx

2. Originally Posted by shanzwhaat
For my homework tonight, there is this one problem that is supposed to be easy but i can't figure it out.

Solve for exact solutions in the interval [0degrees, 360degrees)
[ means include, & ) means exclude
2sinx -1= cscx
I am supposed to find the degree value for x. and i think they want me to use cscx=1/sinx
They do! Then multiply by $\displaystyle \sin x$ and you have a quadratic equation in $\displaystyle \sin x$ which you can solve.

3. Thanks, but i did that and got 0degrees, 180degrees, 30 and 150
but the back of the book said the answers were 90degrees, 210degrees, and 330degrees.. Do you know what I'm doing wrong??/

4. Originally Posted by shanzwhaat
Thanks, but i did that and got 0degrees, 180degrees, 30 and 150
but the back of the book said the answers were 90degrees, 210degrees, and 330degrees.. Do you know what I'm doing wrong??/
If $\displaystyle 2\sin x - 1 = \csc x$ then $\displaystyle 2\sin x - 1 = \frac{1}{\sin x}$ so

$\displaystyle 2 \sin^2 x - \sin x -1 = 0$ or $\displaystyle (2 \sin x + 1)(\sin x -1) = 0\;\;\Rightarrow\;\; \sin x = - \frac{1}{2}, 1$

Did you get that?

5. Originally Posted by shanzwhaat
For my homework tonight, there is this one problem that is supposed to be easy but i can't figure it out.

Solve for exact solutions in the interval [0degrees, 360degrees)
[ means include, & ) means exclude
2sinx -1= cscx
I am supposed to find the degree value for x. and i think they want me to use cscx=1/sinx
$\displaystyle 2\sin{x} - 1 = \csc{x}$

$\displaystyle 2\sin{x} - 1 = \frac{1}{\sin{x}}$

$\displaystyle 2\sin^2{x} - \sin{x} = 1$

$\displaystyle 2\sin^2{x} - \sin{x} - 1 = 0$.

Let $\displaystyle X = \sin{x}$ so

$\displaystyle 2X^2 - X - 1 = 0$.

$\displaystyle 2X^2 - 2X + X - 1 = 0$

$\displaystyle 2X(X - 1) + 1(X - 1) = 0$

$\displaystyle (X - 1)(2X + 1) = 0$

$\displaystyle X - 1 = 0$ or $\displaystyle 2X + 1 = 0$

$\displaystyle X = 1$ or $\displaystyle X = -\frac{1}{2}$

So $\displaystyle \sin{x} = 1$ or $\displaystyle \sin{x} = -\frac{1}{2}$.

In the domain $\displaystyle x \in [0^{\circ}, 360^{\circ})$, $\displaystyle \sin{x} = 1$ if $\displaystyle x = 90^{\circ}$.

In the domain $\displaystyle x \in [0^{\circ}, 360^{\circ}), \sin{x} = -\frac{1}{2}$ if $\displaystyle x = 210^{\circ}$ (Quadrant 3) or $\displaystyle x = 330^{\circ}$ (Quadrant 4).

Does that make sense?

6. YES! thank you everyone who answered!