Practice Test

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

Find the area of the triangle in Figure 1. 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.

Figure 1

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?

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

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

Convert the polar equation to a Cartesian equation: $x^{2} + y^{2} = 5 y. x^{2} + y^{2} = 5 y.$

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

Test the equation for symmetry: $r = - 4 \sin (2 θ). r = - 4 \sin (2 θ).$

Graph $r = 3 + 3 \cos θ . r = 3 + 3 \cos$

10.

Graph $r = 3 - 5 sin θ . r = 3 - 5 sin$

11.

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

12.

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

13.

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

Given $z_{1} = 8 cis (36°) z_{1} = 8 cis (36°)$ and $z_{2} = 2 cis (15°), z_{2} = 2 cis (15°),$ evaluate each expression.

14.

$z_{1} z_{2} z_{1} z_{2}$

15.

$\frac{z_{1}}{z_{2}} \frac{z_{1}}{z_{2}}$

16.

${(z_{2})}^{3} {(z_{2})}^{3}$

17.

$\sqrt{z_{1}} \sqrt{z_{1}}$

18.

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

19.

Eliminate the parameter $t t$ to rewrite the following parametric equations as a Cartesian equation: ${\begin{array}{l} x (t) = t + 1 \\ y (t) = 2 t^{2} \end{array} . {\begin{array}{l} x (t) = t + 1 \\ y (t) = 2 t^{2} \end{array} .$

20.

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

21.

Graph the set of parametric equations and find the Cartesian equation: ${\begin{array}{l} x (t) = - 2 \sin t \\ y (t) = 5 \cos t \end{array} . {\begin{array}{l} x (t) = - 2 \sin \\ y (t) = 5 \cos \end{array} .$

22.

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.

ⓐFind the parametric equations to model the path of the ball.
ⓑWhere is the ball after 2 seconds?

For the following exercises, use the vectors u = i − 3j and v = 2i + 3j.

23.

Find 2u − 3v.

24.

Calculate $u \cdot v . u \cdot v .$

25.

Find a unit vector in the same direction as $v . v .$

26.

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