# A few HW questions I dont get

• Dec 3rd 2006, 02:25 PM
slaying_ice
A few HW questions I dont get
Using the conditions below
sinA=3/4, A lies in quadrant 2
CosB=12/13, B lies in quadrant 1

Find the following:

Cos (A+B) Tan(A-B)

Cos B/2

Solve the equation in th interval [0,2pie)

2cos^2x-3cosx+1

Use Half angle forumlas to find the exact value of the expression

cos 22.5

cos^2 75 - sin^2 75
• Dec 3rd 2006, 04:06 PM
topsquark
Quote:

Originally Posted by slaying_ice
Using the conditions below
sinA=3/4, A lies in quadrant 2
CosB=12/13, B lies in quadrant 1

Find the following:

Cos (A+B) Tan(A-B)

Cos B/2

First:
$\displaystyle cos(A + B) = cos(A) cos(B) - sin(A) sin(B)$

Second:
$\displaystyle tan(A - B) = \frac{sin(A - B)}{cos(A - B)}$

$\displaystyle tan(A - B) = \frac{sin(A)cos(B) - sin(B)cos(A)}{cos(A)cos(B) + sin(A)sin(B)}$

Now:
$\displaystyle sin(A) = \frac{3}{4}$ and is in the 2nd quadrant.

Thus: $\displaystyle cos(A) = -\sqrt{1 - sin^2(A)} = -\frac{1}{2}$

Again:
$\displaystyle cos(B) = \frac{12}{13}$ and is in the first quadrant.

Thus: $\displaystyle sin(B) = +\sqrt{1 - cos^2(B)} = \frac{5}{13}$

To the problems:
$\displaystyle cos(A + B) = cos(A) cos(B) - sin(A) sin(B)$

$\displaystyle cos(A + B) = -\frac{1}{2} \cdot \frac{12}{13} - \frac{3}{4} \cdot \frac{5}{13} = -\frac{3}{4}$

$\displaystyle tan(A - B) = \frac{sin(A)cos(B) - sin(B)cos(A)}{cos(A)cos(B) + sin(A)sin(B)}$

$\displaystyle tan(A - B) = \frac{\frac{3}{4} \cdot \frac{12}{13} - \frac{5}{13} \cdot -\frac{1}{2}}{-\frac{1}{2} \cdot \frac{12}{13} + \frac{3}{4} \cdot \frac{5}{13}} = - \frac{46}{9}$

So
$\displaystyle cos(A + B) \cdot tan(A - B) = -\frac{3}{4} \cdot -\frac{46}{9} = \frac{23}{6}$
================================================== =======
$\displaystyle cos \left ( \frac{B}{2} \right ) = \pm \sqrt{\frac{1 + cos(B)}{2}}$

$\displaystyle cos \left ( \frac{B}{2} \right ) = \pm \sqrt{\frac{1 + \frac{12}{13}}{2}} = \pm \sqrt{\frac{25}{26}} = \pm \frac{5}{\sqrt{26}}$

Since B is in the 1st quadrant, so is B/2. Thus
$\displaystyle cos \left ( \frac{B}{2} \right ) = \frac{5 \sqrt{26}}{26}$

-Dan
• Dec 3rd 2006, 04:11 PM
topsquark
Quote:

Originally Posted by slaying_ice
Solve the equation in th interval [0,2pie)

2cos^2x-3cosx+1

By the way, the Greek letter $\displaystyle \pi$ is spelled "pi" not "pie."

There's no equation to solve here!

I'll assume you meant:
$\displaystyle 2cos^2(x) - 3cos(x) + 1 = 0$

Let's simplify a bit: Let $\displaystyle y = cos(x)$.
The we have:
$\displaystyle 2y^2 - 3y + 1 = 0$

This factors:
$\displaystyle (2y - 1)(y - 1) = 0$

So $\displaystyle y = \frac{1}{2}$ or $\displaystyle y = 1$.

Thus
$\displaystyle cos(x) = \frac{1}{2}$ ==> $\displaystyle x = \frac{\pi}{3}, \frac{5 \pi}{3}$ rad

or

$\displaystyle cos(x) = 1$ ==> $\displaystyle x = 0$.

So the full solution set on $\displaystyle [0, 2 \pi)$ is $\displaystyle x = 0, \frac{\pi}{3}, \frac{5 \pi}{3}$.

-Dan
• Dec 3rd 2006, 04:18 PM
slaying_ice
On the second problem you assumed correct on what the equation was.
• Dec 3rd 2006, 04:20 PM
topsquark
Quote:

Originally Posted by slaying_ice
Use Half angle forumlas to find the exact value of the expression

cos 22.5

cos^2 75 - sin^2 75

The first one:
$\displaystyle cos(22.5) = cos \left ( \frac{45}{2} \right ) = \sqrt{ \frac{1 + cos(45)}{2} } = \sqrt{ \frac{1 + \frac{ \sqrt{2} }{2} }{2} } = \frac{ \sqrt{2 + \sqrt{2} }}{2}$

The second one actually isn't a half-angle formula, it's a double angle formula:
$\displaystyle cos(150) = cos(2 \cdot 75) = cos^2(75) - sin^2(75)$

and
$\displaystyle cos(150) = -cos(30) = -\frac{\sqrt{3}}{2}$

-Dan
• Dec 3rd 2006, 04:22 PM
slaying_ice
The file you uploaded to solve the double angle problem isnt showing up. Plus I have a couple more problems that need to be solved if you dont mind

Find the exact valuse of Sin2x Cos2x and tan 2x given tan=4/3 in the first quadrant.

and

Find the exact valuse of sin x/2 cos x/2 and tan x/2 when cosx= -5/13 and x is in the third quadrant

Thank you

~Kyle
• Dec 3rd 2006, 04:34 PM
topsquark
Quote:

Originally Posted by slaying_ice
The file you uploaded to solve the double angle problem isnt showing up.

I fixed it. Sorry about that.

-Dan
• Dec 3rd 2006, 04:37 PM
slaying_ice
Plus I have a couple more problems that need to be solved if you dont mind

Find the exact valuse of Sin2x Cos2x and tan 2x given tan=4/3 in the first quadrant.

and

Find the exact valuse of sin x/2 cos x/2 and tan x/2 when cosx= -5/13 and x is in the third quadrant

Thank you

~Kyle
• Dec 3rd 2006, 04:51 PM
topsquark
Quote:

Originally Posted by slaying_ice
Plus I have a couple more problems that need to be solved if you dont mind

Find the exact valuse of Sin2x Cos2x and tan 2x given tan=4/3 in the first quadrant.

$\displaystyle tan(x) = \frac{4}{3}$ so

$\displaystyle sin(x) = sin \left ( tan^{-1} \left ( \frac{4}{3} \right ) \right )$

This is a 3 - 4 - 5 triangle, so
$\displaystyle sin(x) = \frac{4}{5}$.

Similarly:
$\displaystyle cos(x) = \frac{3}{5}$

So:
$\displaystyle sin(2x) = 2 sin(x) cos(x) = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} = \frac{24}{25}$

$\displaystyle cos(2x) = cos^2(x) - sin^2(x) = \left (\frac{4}{5} \right ) ^2 - \left ( \frac{3}{5} \right ) ^2 = \frac{7}{25}$

$\displaystyle tan(2x) = \frac{sin(2x)}{cos(2x)} = \frac{ \frac{24}{25} }{ \frac{7}{25} } = \frac{24}{7}$

-Dan
• Dec 3rd 2006, 05:00 PM
topsquark
Quote:

Originally Posted by slaying_ice
Find the exact valuse of sin x/2 cos x/2 and tan x/2 when cosx= -5/13 and x is in the third quadrant

$\displaystyle cos(x) = -\frac{5}{13}$

Thus
$\displaystyle sin \left ( \frac{x}{2} \right ) = \pm \sqrt{\frac{1 - cos(x)}{2}} = \pm \sqrt{ \frac{1 + \frac{5}{13}}{2} } = \pm \frac{3}{\sqrt{13}}$

Thus $\displaystyle sin \left ( \frac{x}{2} \right ) = \frac{3 \sqrt{13}}{13}$

$\displaystyle cos \left ( \frac{x}{2} \right ) = \pm \sqrt{\frac{1 + cos(x)}{2}} = \pm \sqrt{ \frac{1 - \frac{5}{13}}{2} } = \pm \frac{2}{\sqrt{13}}$

Again, since x/2 is 2nd quadrant:
$\displaystyle cos \left ( \frac{x}{2} \right ) = - \frac{2\sqrt{13}}{13}$

$\displaystyle tan \left ( \frac{x}{2} \right ) = \frac{sin \left ( \frac{x}{2} \right )}{cos \left ( \frac{x}{2} \right )} = \frac{ \frac{3 \sqrt{13}}{13} }{ - \frac{2\sqrt{13}}{13} } = - \frac{3}{2}$

-Dan
• Dec 3rd 2006, 05:03 PM
slaying_ice
Thank you very much you have saved me from homework doom
• Dec 4th 2006, 05:07 AM
topsquark
Quote:

Originally Posted by slaying_ice
Thank you very much you have saved me from homework doom

But I LIKE Doom! (Doom, DoomII, etc...)

-Dan