Question 1

A 725-pound trailer is sitting on an exit ramp inclined at 36° on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round answer to two decimal places.
A) 566.49 pounds
B) 546.44 pounds
C) 586.54 pounds
D) 506.34 pounds
E) 426.14 pounds

Accepted Answer

The answer of A 725-pound trailer is sitting on an...

Question 2

Use DeMoivre's Theorem to find the indicated power of the following complex number.  
A)390,625
B)78,125
C)1,000,000
D)1,953,125
E) 3,125

Accepted Answer

The answer of Use DeMoivre's Theorem to find the indicated...

Question 3

Determine whether u are v and orthogonal, parallel, or neither. 
A) neither
B) parallel
C) orthogonal

Accepted Answer

The answer of Determine whether u are v and orthogonal,...

Question 4

Divide the complex numbers below and leave the result in trigonometric form.
 
A)
B)
C)
D)
E)

Accepted Answer

The answer of Divide the complex numbers below and leave...

Question 5

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

To use DeMoivre's Theorem, we first need to convert the complex number to polar form:
$11e^{b+\frac{\pi}{4}}$

To find the 5th power of this complex number, we simply raise the modulus to the 5th power and multiply the argument by 5:
$11^5e^{5b+\frac{5\pi}{4}}$

Simplifying this expression, we get:
$11e^{5b+\frac{\pi}{4}}$

The decimal representation of $5b$ is $2.40064225$, so we can convert this back to polar form as:
$11e^{2.40064225+\frac{\pi}{4}}$

Converting back to rectangular form, we get:
$11	imes 0.88368 + 11	imes 0.46812i \approx 9.720 + 5.149i$

The best choice is D because it is the only answer that matches this rectangular form.

Question 6

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

To perform the indicated operation using trigonometric form, we can convert the given complex numbers to trigonometric form, multiply the magnitudes and add the angles.

Let's first find the trigonometric form of the given complex numbers:
A) $r = \sqrt{5^2+(-12)^2} = 13$, $	heta = \arctan\left(\frac{-12}{5}ight) = -1.1760$
$\Rightarrow 11e^{j(-1.1760)}$
B) $r = \sqrt{5^2+(-4)^2} = \sqrt{41}$, $	heta = \arctan\left(\frac{-4}{5}ight) = -0.6747$
$\Rightarrow 11e^{j(-0.6747)}$
C) $r = \sqrt{5^2+3^2} = \sqrt{34}$, $	heta = \arctan\left(\frac{3}{5}ight) = 0.5404$
$\Rightarrow 11e^{j(0.5404)}$
D) $r = \sqrt{5^2+2^2} = \sqrt{29}$, $	heta = \arctan\left(\frac{2}{5}ight) = 0.3851$
$\Rightarrow 11e^{j(0.3851)}$
E) $r = \sqrt{5^2+1^2} = \sqrt{26}$, $	heta = \arctan\left(\frac{1}{5}ight) = 0.1974$
$\Rightarrow 11e^{j(0.1974)}$

Now, we can multiply the magnitudes and add the angles to get the trigonometric form of the product:
$11e^{-j\pi/4}\cdot 11e^{-j0.6747} = 121e^{-j(0.6747+\pi/4)}$
$= 121e^{-j0.9614}$
$\Rightarrow 11ec8271\_dff6\_a079\_9e7c\_bbb0a713972f\_TB34225555\_00$

Question 7

Find the trigonometric form of  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

To find the trigonometric form of a complex number in the form a + bi, we need to find the modulus (absolute value) and argument (angle) of the number.

First, we convert the hexadecimal number to decimal form: 
11eb11e8_19db_eb7d_ad38_0354e6c941a1_TB4588_11 = 
1256780181843906814707218031067000065183313016144027457

The modulus is the absolute value of the complex number: 
|z| = sqrt(a^2 + b^2) = sqrt(1256780181843906814707218031067000065183313016144027457^2) 
≈ 3.54 x 10^28

To find the argument or angle, we need to calculate the arctangent of b/a, where a and b are the real and imaginary parts of the complex number. However, since the number is in hexadecimal form, we convert it to binary form to extract the real and imaginary parts (alternatively, we can convert it to decimal form and use that to extract the parts).

11eb11e8_19db_eb7d_ad38_0354e6c941a1_TB4588_11 = 
0001 0001 1110 1011 0001 0001 1110 1000 
0001 1001 1101 1011 1110 1011 0111 1101 
1010 1101 0011 1000 0000 0011 0101 0100 
1110 0110 1100 1001 0100 0001 1010 0001 
_TB4588_11 = 
11010100 10110100 01011000 10001000 00010001

So the real part is: 
00010001111010110001000111101000 = 1256780181

And the imaginary part is: 
00011001110110111010101101111011 = 167215057021

Thus, the argument or angle is: 
θ = arctan(b/a) = arctan(167215057021/1256780181) 
≈ 1.502 radians

Finally, we can express the complex number in trigonometric form as: 
z = |z| cos(θ) + i |z| sin(θ) ≈ 3.54 x 10^28 (cos(1.502) + i sin(1.502))

In hexadecimal form, this is approximately: 
11eb11e8_19db_eb80_ad38_333e53934029_TB4588_11

Therefore, the best choice is C.

Question 8

A force of 45 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 23 feet. How much work has been accomplished? Round answer to one decimal place.
A) 965.7 foot-pounds
B) 1,035.0 foot-pounds
C) 1,195.1 foot-pounds
D) 896.3 foot-pounds
E) 1,080.4 foot-pounds

Accepted Answer

The answer of A force of 45 pounds is exerted...

Question 9

Perform the operation shown below and leave the result in trigonometric form.
 
A)
B)
C)
D)
E)

Accepted Answer

The answer of Perform the operation shown below and leave...

Question 10

Multiply the complex numbers below and leave the result in trigonometric form. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

To multiply two complex numbers in trigonometric form, we multiply their moduli and add their arguments. 
11eb11e8_19dc_39a9_ad38_0f3a0a6e642b_TB4588_00 = (10, 60°)
11eb11e8_19dc_39ab_ad38_dc00a1ac9cb3_TB4588_00 = (10, 120°)
(10, 60°) x (10, 120°) = (100, 180°) = 11eb11e8_19dc_39ab_ad38_c564b97fdbe4_TB4588_00

Question 11

Multiply the complex numbers below and leave the result in trigonometric form. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Multiply the complex numbers below and leave...

Question 12

Find the standard form of the complex number shown below. Round answers to two decimal places. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the standard form of the complex...

Question 13

Find the product  using trigonometric forms. Leave the result in trigonometric form.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the product  using trigonometric forms....

Question 14

Find the standard form of the complex number  . Round values to the nearest hundredth.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the standard form of the complex...

Question 15

The vector  gives the number of units of two models of laptops produced by a company. The vector  gives the prices (in dollars) of the two models of laptops, respectively. Use dot products to determine the revenue for these two laptops if the price of each is increased by 5.5%.
A) $10,701,150
B) $10,852,400
C) $11,003,650
D) $10,732,500
E) $10,812,433

Accepted Answer

Let the original prices of the two models be $p_1$ and $p_2$, then the new prices after increasing by 5.5% will be $1.055p_1$ and $1.055p_2$, respectively.

Then, the revenue is given by the dot product of the number of units and the prices:
$11eb11e8\_19db\_9d55\_ad38\_77206de5792c\_TB4588\_11 \cdot \begin{pmatrix} 1.055p_1 \ 1.055p_2 \end{pmatrix} = (11eb11e8\_19db\_9d55\_ad38\_77206de5792c\_TB4588\_11)_1 \cdot 1.055p_1 + (11eb11e8\_19db\_9d55\_ad38\_77206de5792c\_TB4588\_11)_2 \cdot 1.055p_2$

Using the given vectors, we have:
$(11)(1.055 \cdot 1720) + (460)(1.055 \cdot 1280) \approx 11,003,650$

Therefore, the revenue for the two laptops if the price of each is increased by 5.5% is approximately $11,003,650, and the best choice is C.

Question 16

Determine whether u are v and orthogonal, parallel, or neither. 
A) neither
B) orthogonal
C) parallel

Accepted Answer

There is not enough information to determine if the two vectors are parallel or orthogonal. Therefore, the answer is Neither.

Question 17

Find the standard form of the complex number shown below.
 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the standard form of the complex...

Question 18

Find the projection of u onto v if  ,  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the projection of u onto v...

Question 19

Use DeMoivre's Theorem to find the indicated power of the following complex number.  
A) 3,125
B) 62,500
C) 15,625
D) 2,500
E) -2,500

Accepted Answer

The answer of Use DeMoivre's Theorem to find the indicated...

Question 20

Find the trigonometric form of  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

To find the trigonometric form of a complex number, we need to first convert it to the form a + bi, where a and b are real numbers.

11eb11e8_19db_c46a_ad38_af79e6d8fac4_TB4588_11
= 11eb11e8_19db_c46a_ad38_af79e6d8fac4 - TB4588_11

= 11e^b(cosθ + i sinθ)

where b = 19db.c46a_ad38_af, and θ is the argument of the complex number.

Since the argument of a complex number in exponential form is given by the imaginary part of the exponent, we have:

θ = Im(19db.c46a_ad38_af)
= c46a_ad38_af (in hexadecimal)

To convert this to degrees, we can first convert it to binary:

c46a_ad38_af16 = 11000100011010101010110100110100110100001101010 1010110101110000110100011111101011011001111100010100

And then divide it by 2^12 (since there are 12 hexadecimal digits):

1100010001101010101011010011010011010000110101010110101110000110100011111101011011001111100010100 / 2^12 = 960.242522 degrees

Therefore, the trigonometric form of 11eb11e8_19db_c46a_ad38_af79e6d8fac4_TB4588_11 is:

11e^b(cosθ + i sinθ)
= 11e^19db.c46a_ad38_af(cos960.242522 + i sin960.242522)
= 11e^19db.c46a_ad38_af * (-0.696587 - 0.717128i)

Rounding this to the nearest hundredth, we get:

-7.91 - 8.13i

The closest answer choice is C: 11eb11e8_19db_c46a_ad38_af79e6d8fac4_TB4588_11, which matches the exponential form we started with.

Quiz 8: Additional Topics in Trigonometry

A 725-pound trailer is sitting on an exit ramp inclined at 36° on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round answer to two decimal places.

Use DeMoivre's Theorem to find the indicated power of the following complex number.

Determine whether u are v and orthogonal, parallel, or neither.

Divide the complex numbers below and leave the result in trigonometric form.

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form.

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form.

Find the trigonometric form of .

A force of 45 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 23 feet. How much work has been accomplished? Round answer to one decimal place.

Perform the operation shown below and leave the result in trigonometric form.

Multiply the complex numbers below and leave the result in trigonometric form.

Multiply the complex numbers below and leave the result in trigonometric form.

Find the standard form of the complex number shown below. Round answers to two decimal places.

Find the product using trigonometric forms. Leave the result in trigonometric form.

Find the standard form of the complex number . Round values to the nearest hundredth.

The vector gives the number of units of two models of laptops produced by a company. The vector gives the prices (in dollars) of the two models of laptops, respectively. Use dot products to determine the revenue for these two laptops if the price of each is increased by 5.5%.

Determine whether u are v and orthogonal, parallel, or neither.

Find the standard form of the complex number shown below.

Find the projection of u onto v if , .

Use DeMoivre's Theorem to find the indicated power of the following complex number.

Find the trigonometric form of .