1. ## New to trigonometry

Ok, I've been looking around on wikipedia etc. but I really don't know how to find the answer to this question at all... would be great with some input.

There are two values that fulfil $sin\theta=\frac{\sqrt{3}}{2}$

they are $\theta=\:\:,$

second part of the question:

for $sin2\theta=\frac{\sqrt{3}}{2}$ there are 4 values,

if $\theta=\theta_1,\:\:\theta_2,\:\:\theta_3,\:\:\the ta_4$ then

$\theta_3-\theta_1=\theta_4-\theta_2=$

and $\theta_2-\theta_1=\theta_4-\theta_3=$

thanks for any help.

2. Originally Posted by davidman
Ok, I've been looking around on wikipedia etc. but I really don't know how to find the answer to this question at all... would be great with some input.

There are two values that fulfil $sin\theta=\frac{\sqrt{3}}{2}$

they are $\theta=\:\:,$

second part of the question:

for $sin2\theta=\frac{\sqrt{3}}{2}$ there are 4 values,

if $\theta=\theta_1,\:\:\theta_2,\:\:\theta_3,\:\:\the ta_4$ then

$\theta_3-\theta_1=\theta_4-\theta_2=$

and $\theta_2-\theta_1=\theta_4-\theta_3=$

thanks for any help.
First, draw yourself an equilateral triangle of side length $2$ units.

Then draw a straight line perpendicular to the base and going through the "top" angle.

You will now have two $30^\circ, 60^\circ, 90^\circ$ triangles.

Their dimensions are $1, \sqrt{3}, 2$ (you can check this using Pythagoras).

Remember that $\sin{\theta} = \frac{\textrm{Opposite}}{\textrm{Hypotenuse}}$.

Which angle (out of $30^\circ$ and $60^\circ$) has the Opposite side with length $\sqrt{3}$ units and the Hypotenuse as $2$ units?

3. Originally Posted by davidman
Ok, I've been looking around on wikipedia etc. but I really don't know how to find the answer to this question at all... would be great with some input.

There are two values that fulfil $sin\theta=\frac{\sqrt{3}}{2}$

they are $\theta=\:\:,$

second part of the question:

for $sin2\theta=\frac{\sqrt{3}}{2}$ there are 4 values,

if $\theta=\theta_1,\:\:\theta_2,\:\:\theta_3,\:\:\the ta_4$ then

$\theta_3-\theta_1=\theta_4-\theta_2=$

and $\theta_2-\theta_1=\theta_4-\theta_3=$

thanks for any help.
I think that you mean to say that $\sin\theta=\frac{\sqrt{3}}{2}$ has two solutions for $0\leq\theta\leq2\pi$ ? And you wish to find the solutions?

Well, $\sqrt3/2$ is given by evaluating $\sin\theta$ at what is commonly referred to as one of the "special angles". Unfortunately, I can't draw too well on this forum, but here's something for you to look at:

Math Concepts Explained: Special Angles in Trigonometry

4. ## Q1,q2

Originally Posted by davidman
Ok, I've been looking around on wikipedia etc. but I really don't know how to find the answer to this question at all... would be great with some input.

There are two values that fulfil $sin\theta=\frac{\sqrt{3}}{2}$

they are $\theta=\:\:,$

second part of the question:

for $sin2\theta=\frac{\sqrt{3}}{2}$ there are 4 values,

if $\theta=\theta_1,\:\:\theta_2,\:\:\theta_3,\:\:\the ta_4$ then

$\theta_3-\theta_1=\theta_4-\theta_2=$

and $\theta_2-\theta_1=\theta_4-\theta_3=$

thanks for any help.
$sin\theta=\frac{\sqrt{3}}{2}$

$\frac{\pi}{3},\frac{2\pi}{3}$

these are quadrants 1 and 2 respectfully

5. Originally Posted by Prove It
Which angle (out of $30^\circ$ and $60^\circ$) has the Opposite side with length $\sqrt{3}$ units and the Hypotenuse as $2$ units?
$sin60=\frac{\sqrt{3}}{2}\:\:sin30=\frac{1}{2}\:\:s in90=1$

Thanks!

Originally Posted by VonNemo19
I think that you mean to say that $\sin\theta=\frac{\sqrt{3}}{2}$ has two solutions for $0\leq\theta\leq2\pi$ ? And you wish to find the solutions?
Exactly that, yes.

Originally Posted by bigwave
$sin\theta=\frac{\sqrt{3}}{2}$

$\frac{\pi}{3},\frac{2\pi}{3}$

these are quadrants 1 and 2 respectfully
this brings me to my next question... how do you get two answers when I only get one from evaluating triangles with "special angles"? And how to go from degrees to radian... ?

6. you may be looking at a triangle in just one quadrant

see the attached very familiar demonstration of "special angles"

$(\cos\theta,\sin\theta)$

7. OK, now that you have found the "focus angle", we now go to a unit circle approach.

A unit circle is a circle of radius 1.

Have a look at the picture below:

For any angle made with the positive $x$ axis (call it $\theta$), the vertical distance is $\sin{\theta}$ and the horizontal distance is $\cos{\theta}$.

If your angle is $60^\circ = \frac{\pi}{3}^C$, can you see another angle which has the same vertical distance?

8. Ok, I see how there would be two solutions. Thinking in degrees, $180-\theta=\theta_2$

however, about the second half of the original question,

$sin2\theta=\frac{\sqrt{3}}{2}$

considering the first part of the question, $2\theta=60\:\:, \theta=30$ makes sense the way I see it. But how are there four solutions and not two?

9. radians x $\left(\frac{180^o}{\pi}\right) =$ degrees

degrees x $\left(\frac{\pi}{180^o}\right)=$ radians

10. Originally Posted by davidman
Ok, I see how there would be two solutions. Thinking in degrees, $180-\theta=\theta_2$

however, about the second half of the original question,

$sin2\theta=\frac{\sqrt{3}}{2}$

considering the first part of the question, $2\theta=60\:\:, \theta=30$ makes sense the way I see it. But how are there four solutions and not two?
OK, for any given domain, the number of solutions depends on the period of the function.

Can you see that if you go around the circle (i.e. if you travel $360^\circ$) you get to the same angle?

If you travel another $360^\circ$, you get back to the same angle again.

And so on, and so on...

So if you wanted to show all solutions to

$\sin{x} = \frac{\sqrt{3}}{2}$

then $x = \left\{ 60^\circ, 120^\circ \right \} + 360^\circ n$, where $n$ is some integer representing the number of times you have gone around the unit circle.

Now if $x = 2\theta$ then

$2\theta = \left \{ 60^\circ, 120^\circ \right\} + 360^\circ n$.

Now, to solve for $\theta$ we have to divide both sides of the equation by $2$. This includes the $360^\circ$...

So $\theta = \left \{ 30^\circ, 60^\circ \right\} + 180^\circ n$.

So letting $n = 0$ we have

$\theta = \left \{ 30^\circ, 60^\circ \right \}$.

If we let $n = 1$ we have

$\theta = \left \{ 210^\circ, 240^\circ \right \}$.

If we let $n = 2$ we have

$\theta = \left \{ 390^\circ, 420^\circ \right \}$. These solutions are out of the original $360^\circ$.

So the four solutions are

$\theta = \left \{ 30^\circ, 60^\circ, 210^\circ, 240^\circ \right \}$.

11. Originally Posted by Prove It
Awesome explanation.
Ooooooooohhhhhhhhhh, I get it now!

$\theta_1=\frac{\pi}{6}\:,\:\theta_2=\frac{\pi}{3}\ :,\:\theta_3=\frac{7}{6}\pi\:,\:\theta_4=\frac{12} {9}\pi$