Jay Abramson

Review Exercises

Non-right Triangles: Law of Sines

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

1.

$β = 50°, a = 105, b = 45$

2.

$α = 43.1°, a = 184.2, b = 242.8$

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 $A$ and the elevation of the plane.

Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.

Figure 1

Non-right Triangles: Law of Cosines

6.

Solve the triangle, rounding to the nearest tenth, assuming $α$ is opposite side $a, β$ is opposite side $b,$ and $γ$ is opposite side $c : a = 4, b = 6, c = 8.$

7.

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

A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.

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.

Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.

Figure 3

Polar Coordinates

10.

Plot the point with polar coordinates $(3, \frac{π}{6}) .$

11.

Plot the point with polar coordinates $(5, - \frac{2 π}{3})$

12.

Convert $(6, - \frac{3 π}{4})$ to rectangular coordinates.

13.

Convert $(- 2, \frac{3 π}{2})$ to rectangular coordinates.

14.

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

15.

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

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

16.

$x = - 2$

17.

$x^{2} + y^{2} = 64$

18.

$x^{2} + y^{2} = - 2 y$

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

19.

$r = 7 cos θ$

20.

$r = \frac{- 2}{4 \cos θ + \sin θ}$

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

21.

$θ = \frac{3 π}{4}$

22.

$r = 5 \sec θ$

Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

23.

$r = 4 + 4 \sin θ$

24.

$r = 7$

25.

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

26.

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

27.

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

Polar Form of Complex Numbers

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

28.

$- 2 + 6 i$

29.

$4 - 3 i$

Write the complex number in polar form.

30.

$5 + 9 i$

31.

$\frac{1}{2} - \frac{\sqrt{3}}{2} i$

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

32.

$z = 5 cis (\frac{5 π}{6})$

33.

$z = 3 cis (40°)$

For the following exercises, find the product $z_{1} z_{2}$ in polar form.

34.

$z_{1} = 2 cis (89°)$

$z_{2} = 5 cis (23°)$

35.

$z_{1} = 10 cis (\frac{π}{6})$

$z_{2} = 6 cis (\frac{π}{3})$

For the following exercises, find the quotient $\frac{z_{1}}{z_{2}}$ in polar form.

36.

$z_{1} = 12 cis (55°)$

$z_{2} = 3 cis (18°)$

37.

$z_{1} = 27 cis (\frac{5 π}{3})$

$z_{2} = 9 cis (\frac{π}{3})$

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

38.

Find $z^{4}$ when $z = 2 cis (70°)$

39.

Find $z^{2}$ when $z = 5 cis (\frac{3 π}{4})$

For the following exercises, evaluate each root.

40.

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

41.

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

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

42.

$6 - 2 i$

43.

$- 1 + 3 i$

Parametric Equations

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

44.

${\begin{array}{l} x (t) = 3 t - 1 \\ y (t) = \sqrt{t} \end{array}$

45.

${\begin{cases} x (t) = - \cos t \\ y (t) = 2 \sin^{2} t \end{cases}$

46.

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

47.

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

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.

${\begin{array}{l} x (t) = 3 t^{2} \\ y (t) = 2 t - 1 \end{array}$

49.

${\begin{array}{l} x (t) = e^{t} \\ y (t) = - 2 e^{5 t} \end{array}$

50.

${\begin{array}{l} x (t) = 3 \cos t \\ y (t) = 2 \sin t \end{array}$

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.

ⓐ Find the parametric equations to model the path of the ball.
ⓑ Where is the ball after 3 seconds?

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 $P_{4} .$

52.

$P_{1} = (- 1, 4), P_{2} = (3, 1), P_{3} = (5, 5)$ and $P_{4} = (9, 2)$

53.

$P_{1} = (6, 11), P_{2} = (- 2, 8), P_{3} = (0, - 1)$ and $P_{4} = (- 8, 2)$

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.

54.

u − v

55.

2v − u + w

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

56.

a = 8i − 6j

57.

b = −3i − j

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

58.

$〈 6, −2 〉$

59.

$〈 −3, −3 〉$

For the following exercises, calculate $u \cdot v .$

60.

u = −2i + j and v = 3i + 7j

61.

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

62.

Given v $= 〈 −3, 4 〉$ draw v, 2v, and $\frac{1}{2}$ v.

63.

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

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.

Figure 4

64.

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.