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For the following exercises, evaluate each root.

Evaluate the cube root of z when z = 64 cis ( 210° ) .

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Evaluate the square root of z when z = 25 cis ( 3 π 2 ) .

5 cis ( 3 π 4 ) , 5 cis ( 7 π 4 )

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For the following exercises, plot the complex number in the complex plane.

Parametric Equations

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation.

{ x ( t ) = 3 t 1 y ( t ) = t

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{ x ( t ) = cos t y ( t ) = 2 sin 2 t  

x 2 + 1 2 y = 1

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Parameterize (write a parametric equation for) each Cartesian equation by using x ( t ) = a cos t and y ( t ) = b sin t for x 2 25 + y 2 16 = 1.

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Parameterize the line from ( 2 , 3 ) to ( 4 , 7 ) so that the line is at ( 2 , 3 ) at t = 0 and ( 4 , 7 ) at t = 1.

{ x ( t ) = 2 + 6 t y ( t ) = 3 + 4 t

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Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

{ x ( t ) = 3 t 2 y ( t ) = 2 t 1

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{ x ( t ) = e t y ( t ) = 2 e 5 t

y = 2 x 5

Plot of the given parametric equations.
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{ x ( t ) = 3 cos t y ( t ) = 2 sin t

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A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 3 seconds?
  3. How long is the ball in the air?
  1. { x ( t ) = ( 80 cos ( 40° ) ) t y ( t ) = 16 t 2 + ( 80 sin ( 40° ) ) t + 4
  2. The ball is 14 feet high and 184 feet from where it was launched.
  3. 3.3 seconds
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Vectors

For the following exercises, determine whether the two vectors, u and v , are equal, where u has an initial point P 1 and a terminal point P 2 , and v has an initial point P 3 and a terminal point P 4 .

P 1 = ( 1 , 4 ) , P 2 = ( 3 , 1 ) , P 3 = ( 5 , 5 ) and P 4 = ( 9 , 2 )

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P 1 = ( 6 , 11 ) , P 2 = ( 2 , 8 ) , P 3 = ( 0 , 1 ) and P 4 = ( 8 , 2 )

not equal

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For the following exercises, use the vectors u = 2 i j , v = 4 i 3 j , and w = 2 i + 5 j to evaluate the expression.

For the following exercises, find a unit vector in the same direction as the given vector.

b = −3 i j

3 10 10 i 10 10 j

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For the following exercises, find the magnitude and direction of the vector.

−3 , −3

Magnitude: 3 2 , Direction: 225°

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For the following exercises, calculate u v .

u = −2 i + j and v = 3 i + 7 j

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u = i + 4 j and v = 4 i + 3 j

16

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Given v = −3 , 4 draw v , 2 v , and 1 2 v .

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Given the vectors shown in [link] , sketch u + v , u v and 3 v .

Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude.


Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).

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Given initial point P 1 = ( 3 , 2 ) and terminal point P 2 = ( 5 , 1 ) , write the vector v in terms of i and j . Draw the points and the vector on the graph.

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Practice test

Assume α is opposite side a , β is opposite side b , and γ is opposite side c . Solve the triangle, if possible, and round each answer to the nearest tenth, given β = 68° , b = 21 , c = 16.

α = 67.1° , γ = 44.9° , a = 20.9

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Find the area of the triangle in [link] . Round each answer to the nearest tenth.

A triangle. One angle is 60 degrees with opposite side 6.25. The other two sides are 5 and 7.
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A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

1712 miles

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Convert ( 2 , 2 ) to polar coordinates, and then plot the point.

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Convert ( 2 , π 3 ) to rectangular coordinates.

( 1 , 3 )

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Convert the polar equation to a Cartesian equation: x 2 + y 2 = 5 y.

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Convert to rectangular form and graph: r = 3 csc θ .

y = 3

Plot of the given equation in rectangular form - line y=-3.
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Test the equation for symmetry: r = 4 sin ( 2 θ ).

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Graph r = 3 + 3 cos θ .


Graph of the given equations - a cardioid.

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Graph r = 3 5 sin θ .

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Find the absolute value of the complex number 5 9 i .

106

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Write the complex number in polar form: 4 + i .

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Convert the complex number from polar to rectangular form: z = 5 cis ( 2 π 3 ) .

5 2 + i 5 3 2

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Given z 1 = 8 cis ( 36° ) and z 2 = 2 cis ( 15° ) , evaluate each expression.

z 1

2 2 cis ( 18° ) , 2 2 cis ( 198° )

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Plot the complex number −5 i in the complex plane.

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Eliminate the parameter t to rewrite the following parametric equations as a Cartesian equation: { x ( t ) = t + 1 y ( t ) = 2 t 2 .

y = 2 ( x 1 ) 2

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Parameterize (write a parametric equation for) the following Cartesian equation by using x ( t ) = a cos t and y ( t ) = b sin t : x 2 36 + y 2 100 = 1.

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Graph the set of parametric equations and find the Cartesian equation: { x ( t ) = 2 sin t y ( t ) = 5 cos t .


Graph of the given equations - a vertical ellipse.

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A ball is launched with an initial velocity of 95 feet per second at an angle of 52° to the horizontal. The ball is released at a height of 3.5 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 2 seconds?
  3. How long is the ball in the air?
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For the following exercises, use the vectors u = i − 3 j and v = 2 i + 3 j .

Find 2 u − 3 v .

−4 i − 15 j

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Find a unit vector in the same direction as v .

2 13 13 i + 3 13 13 j

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Given vector v has an initial point P 1 = ( 2 , 2 ) and terminal point P 2 = ( 1 , 0 ) , write the vector v in terms of i and j . On the graph, draw v , and v .

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Questions & Answers

In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
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the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
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100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
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10y
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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