geno3141

For the first problem:  The absolute value of a complex number is the distance from the origin to that point.

So, if you draw a circle whose center is the origin and whose radius is π, every point on that circle will solve the problem.

The two obvious answers are at x = π and x = -π, but you aren't allowed to use them; so choose any other location on the circle.

If you choose the angle 45°, θ= 45° and r = π,

the value of   z = r[cos(θ) + i ·sin(θ)]  becomes  π[cos(45°) + i · sin(45°)]  =  √2/2·π + √2/2·π·i

4x² - 12x - 72  =  0

To reduce the size ofthe numers, divide all numbers by 4:

x² - 3x - 18  =  0

Factor:  (x - 6)(x + 3)  =  0

To solve:  either        x - 6  =  0         or         x + 3  =  0

                                       x  =  6          or               x  =  -3

To simplify  (1 + 3i) / (1 - 3i)  multiply both the numerator and the denominator by the conjugate of the denominator.

The conjugate of  1 - 3i  is  1 + 3i.

So, you need to simplify:  [ (1 + 3i)(1 + 3i) ] / [ (1 - 3i)(1 + 3i) ]     Use FOIL.

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The absolute value of a complex number is the distance that the point, when graphed is from the origin.

Label the point  x + yi  and use the Pytharoean Theorem:  r²  =  x² +  y²

When  x = 7  and  y = 7, what is the value of r?

When  x = 7  and  y = -7, what is the value of r?

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By √4(1 + i), do you mean 2(1 + i)?  If so, use the distributive law.

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By  "^4√4", do you mean the fourth root of 4?  4^(1/4)  =  (2^2)^(1/4)  =  2^(1/2)  =  √2.

√2(1 - i)  =  √2 - √2i

The Pythagorean Theorem:  In a right triangle whose legs have lengths  a  and  b  and whose hypotenuse has length  c,  this formula is true:  a² + b²  =  c².

If the problem is x  =  5/(3x) - 2:

Multiply each term by the denominator, 3x, to get:  3x² = 5 - 6x

Get all the values to one side:                    3x² + 6x - 5  =  0

Factoring won't work, so use the quadratic formula with  a = 3, b = 6 and c = -5.

If the problem is:  x  =  5/(3x - 2):

Multiply each side by  3x -2  to get:  x(3x - 2)  =  5

Multiply out:                                        3x² - 2x  =  5

Get all the values to one side:       3x² - 2x - 5  =  0

Factoring:                                 (3x - 5)(x + 1)  =  0

So, either          3x - 5  =  0          or         x + 1  =  0

                                3x  =  5                            x  =  -1

                                x  = 5/3

We'll need more information.

I don't know what 'FL and 'FR' represent; however,

The reference angle of 195° 30' 32" is also 15° 30' 32", so the cosine of the first angle will have the same absolute value of the cosine of the second angle. Since cosines represent horizontal distances, one angle is as far from the origin to the right as the other is to the left. The center between the two points will be the origin. The average horizontal value will be zero.

I'm dazzled by this -- it can be solved by graphing.

You can make the substitution  1 - 2sin²(x)  =  cos(2x)

to get:  4cos(x)cos²(x-1) + cos(2x)  =  0,

but ...

Your numerator: 15*26997 - 480*839  =  2235

Your denominator: √( [15*15586 - 480²][15*47721 - 839²] )

                        =  √( [3390][11894] )    =  √( 40320660 )  =  6349.855

Final value:  2235 / 6349.855  =  0.35

My argument for #22:

As x→0+, ln(x) → -∞.

When does tan(x) → -∞?

As x → -π/2 (from the plus side), tan(x) → -∞.

So, as tan(x) → -∞ (from the plus side), x → -π/2 

So, as x → -∞, invTan(x) →  -π/2 

So, the limit is  -π/2