Algebra and Trigonometry

# Review Exercises

Algebra and TrigonometryReview Exercises

#### Non-right Triangles: Law of Sines

For the following exercises, assume$α α$is opposite side$a,β a,β$is opposite side$b, b,$and$γ γ$is opposite side$c. c.$Solve each triangle, if possible. Round each answer to the nearest tenth.

1.

$β=50°,a=105,b=45 β=50°,a=105,b=45$

2.

$α=43.1°,a=184.2,b=242.8 α=43.1°,a=184.2,b=242.8$

3.

Solve the triangle.

4.

Find the area of the triangle.

5.

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in Figure 1. Find the distance of the plane from point$A A$and the elevation of the plane.

Figure 1

#### Non-right Triangles: Law of Cosines

6.

Solve the triangle, rounding to the nearest tenth, assuming$α α$is opposite side$a,β a,β$is opposite side$b, b,$and$γ γ$s opposite side

7.

Solve the triangle in Figure 2, rounding to the nearest tenth.

Figure 2
8.

Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

9.

To find the distance between two cities, a satellite calculates the distances and angle shown in Figure 3 (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

Figure 3

#### Polar Coordinates

10.

Plot the point with polar coordinates$( 3, π 6 ). ( 3, π 6 ).$

11.

Plot the point with polar coordinates$( 5,− 2π 3 ) ( 5,− 2π 3 )$

12.

Convert$( 6,− 3π 4 ) ( 6,− 3π 4 )$to rectangular coordinates.

13.

Convert$( −2, 3π 2 ) ( −2, 3π 2 )$to rectangular coordinates.

14.

Convert$( 7,−2 ) ( 7,−2 )$to polar coordinates.

15.

Convert$( −9,−4 ) ( −9,−4 )$ to polar coordinates.

For the following exercises, convert the given Cartesian equation to a polar equation.

16.

$x=−2 x=−2$

17.

$x 2 + y 2 =64 x 2 + y 2 =64$

18.

$x 2 + y 2 =−2y x 2 + y 2 =−2y$

For the following exercises, convert the given polar equation to a Cartesian equation.

19.

$r=7cos θ r=7cos θ$

20.

$r= −2 4cos θ+sin θ r= −2 4cos θ+sin θ$

For the following exercises, convert to rectangular form and graph.

21.

$θ= 3π 4 θ= 3π 4$

22.

$r=5sec θ r=5sec θ$

#### Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

23.

$r=4+4sin θ r=4+4sin θ$

24.

$r=7 r=7$

25.

Sketch a graph of the polar equation$r=1−5sin θ. r=1−5sin θ.$Label the axis intercepts.

26.

Sketch a graph of the polar equation$r=5sin( 7θ ). r=5sin( 7θ ).$

27.

Sketch a graph of the polar equation$r=3−3cos θ r=3−3cos θ$

#### Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.

28.

$−2+6i −2+6i$

29.

$4−​3i 4−​3i$

Write the complex number in polar form.

30.

$5+9i 5+9i$

31.

$1 2 − 3 2 ​i 1 2 − 3 2 ​i$

For the following exercises, convert the complex number from polar to rectangular form.

32.

$z=5cis( 5π 6 ) z=5cis( 5π 6 )$

33.

$z=3cis( 40° ) z=3cis( 40° )$

For the following exercises, find the product$z 1 z 2 z 1 z 2$in polar form.

34.

$z 1 =2cis( 89° ) z 1 =2cis( 89° )$

$z 2 =5cis( 23° ) z 2 =5cis( 23° )$

35.

$z 1 =10cis( π 6 ) z 1 =10cis( π 6 )$

$z 2 =6cis( π 3 ) z 2 =6cis( π 3 )$

For the following exercises, find the quotient$z 1 z 2 z 1 z 2$in polar form.

36.

$z 1 =12cis( 55° ) z 1 =12cis( 55° )$

$z 2 =3cis( 18° ) z 2 =3cis( 18° )$

37.

$z 1 =27cis( 5π 3 ) z 1 =27cis( 5π 3 )$

$z 2 =9cis( π 3 ) z 2 =9cis( π 3 )$

For the following exercises, find the powers of each complex number in polar form.

38.

Find$z 4 z 4$when$z=2cis( 70° ) z=2cis( 70° )$

39.

Find$z 2 z 2$when$z=5cis( 3π 4 ) z=5cis( 3π 4 )$

For the following exercises, evaluate each root.

40.

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

41.

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

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

42.

$6−2i 6−2i$

43.

$−1+3i −1+3i$

#### Parametric Equations

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

44.

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

45.

46.

Parameterize (write a parametric equation for) each Cartesian equation by using$x( t )=acos t x( t )=acos t$and$y(t)=bsin t y(t)=bsin t$for$x 2 25 + y 2 16 =1. x 2 25 + y 2 16 =1.$

47.

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

#### 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.

48.

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

49.

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

50.

${ x(t)=3cos t y(t)=2sin t { x(t)=3cos t y(t)=2sin t$

51.

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?

#### Vectors

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

52.

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

53.

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

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

54.

uv

55.

2vu + w

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

56.

a = 8i − 6j

57.

b = −3ij

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

58.

$〈 6,−2 〉 〈 6,−2 〉$

59.

$〈 −3,−3 〉 〈 −3,−3 〉$

For the following exercises, calculate$u⋅v. u⋅v.$

60.

u = −2i + j and v = 3i + 7j

61.

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

62.

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

63.

Given the vectors shown in Figure 4, sketch u + v, uv and 3v.

Figure 4
64.

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