Solve Trig Expressions: Inverse Sin and Tan -1/2, Inverse Tan 3/4

[bdel89]

Homework Statement

find the exact value of each expression

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

Homework Equations

i know what the answer is because its in the back of the book but i don't know how to solve it and i asked the teacher and he doesn't help anyone.

The Attempt at a Solution

the answer is -48+25√3/39

 

[malawi_glenn]

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

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

 

[bdel89]

formula for tan(a-b) is tan a - tan b/1+ tan a*tan b. i don't know the answer for the first one

 

[malawi_glenn]

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

 

[malawi_glenn]

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

 

[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

 

[Lunar Guy]

Homework Statement

find the exact value of each expression

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

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. :)

 

[malawi_glenn]

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

 

[Lunar Guy]

Yes, that is true because:

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

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