# Analytic trigonometry

1. Feb 10, 2008

### bdel89

1. The problem statement, all variables and given/known data
find the exact value of each expression

tan[inverse sin(-1/2) - inverse tan 3/4]

2. Relevant equations

i know what the answer is because its in the back of the book but i dont know how to solve it and i asked the teacher and he doesnt help anyone.

3. The attempt at a solution

Last edited: Feb 10, 2008
2. Feb 10, 2008

### malawi_glenn

what angle x do you need to get sin(x) = -1/2 ?

What is the forumula for tan(a- b) ?

Last edited: Feb 10, 2008
3. Feb 10, 2008

### bdel89

formula for tan(a-b) is tan a - tan b/1+ tan a*tan b. i dont know the answer for the first one

4. Feb 10, 2008

### malawi_glenn

then you must draw the unit circle, and to "algebraic" trigonometry to find the angle that gives you sin(x) = -1/2 ;)

5. Feb 10, 2008

### malawi_glenn

tan[inverse sin(-1/2) - inverse tan 3/4] is not equal to 48+25√3/39

6. Feb 11, 2008

### TheoMcCloskey

bdel89 - be careful with your expression and use parens' "(" and ")" appropriately to avoid confusion.

Also, you need to familarize yourself with some basic trig relationship for some standard angles (30degree, 60degree, 45 degree). These are usually considered "basic" as the trig function evaluation appear quite often and they are important enough to memorize. If you don't know them, you should review.

Anyway, even if you don't know ArcSin (-1/2), do as malawi_glenn suggests and draw on a unit circle an angle whose Sin is -1/2. From there, compute what the tangent of that angle is. Use this value in your formulae for the tangent of the difference of angles.

By the way, one answer is

-(48+25√3)/39

Last edited: Feb 11, 2008
7. Feb 11, 2008

### Lunar Guy

Okay, let's work inside of the parentheses first:

$$Sin^{-1} -\frac{1}{2}$$

You can look at this as:

$$sin \theta = -\frac{1}{2}$$

$$\theta$$ = -30° or 330°

Convert that into radians, and you get:

$$\frac{11\pi}{6}$$ or $$-\frac{\pi}{6}$$

In order for you to get sin x = -1/2, draw a graph:

_______
\______|
_\_____|
__\____|
___\___|
____\__|
_____\_|
______\|

(Sorry the graph isn't necessarily to scale.)

Where the red line is 2 and the blue line is -1. (rise/run)

Use the pythagorean theorem, and you find out that the purple line = $$\sqrt{3}$$.

Hmm... A 30-60-90 triangle...

Thus, -1 corresponds to the 30° angle. However, the angle is under the x-axis, so the angle is negative. (-30°).

Now, if you use the techniques above, you can easily find the value of:

$$Tan^{-1} \frac{3}{4}$$

...Although you might need to use basic trigonometry, the Law of Sines, or the Law of Cosines to find the angle you need. (Basic trig would be the easiest and quickest to use.)

Once you've found the value of the angle of:

$$Tan^{-1} \frac{3}{4}$$

...You have to use the formula for tan(α-β). Remember to simplify the equation, i.e. getting rid of redicals in the denominator, and you're all set. :)

8. Feb 11, 2008

### malawi_glenn

one does not need the value of arctan(3/4)...

9. Feb 14, 2008

### Lunar Guy

Yes, that is true because:

$$tan[Tan^{-1} (\frac{3}{4}) ]$$

Cancel each other out. Hint, hint... ;)