Algebra and Trigonometry

# Practice Test

1.

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$β=68°,b=21,c=16. β=68°,b=21,c=16.$

2.

Find the area of the triangle in Figure 1. Round each answer to the nearest tenth.

Figure 1
3.

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?

4.

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

5.

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

6.

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

7.

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

8.

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

9.

Graph$r=3+3cos θ. r=3+3cos θ.$

10.

Graph$r=3−5sin θ. r=3−5sin θ.$

11.

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

12.

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

13.

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

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

14.

$z 1 z 2 z 1 z 2$

15.

$z 1 z 2 z 1 z 2$

16.

$( z 2 ) 3 ( z 2 ) 3$

17.

$z 1 z 1$

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: ${ x(t)=t+1 y(t)=2 t 2 . { x(t)=t+1 y(t)=2 t 2 .$

20.

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

21.

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

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.

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?

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

23.

Find 2u − 3v.

24.

Calculate$u⋅v. u⋅v.$

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$−v. −v.$