Advanced Algebra & Trig

Solutions to the Review 2008

 

 

1.      Given the following equation, , find the period and amplitude.   Period is 180 and amplitude is 3.

2.      Write an equation of the cosine function with amplitude 2, period , and phase shift   y = 2cos(2x-180)

3.      Using the general trigonometric equation , what expression is used to determine the amplitude?  period?  vertical shift?   phase shift?

  1.    period
  2.    amplitude
  3.    phase shift
  4.    vertical motion

4.      Evaluate .   This means in the 1st and 2nd quadrants only (principal values for cosine), when is Cosine equal to 1?  This occurs at 0°.

5.      Find the values of  in the interval  that satisfy the equation .  This would happen at 60° and 120°.  This is NOT a principal values problem.

6.      Find the values of  for which the equation .  Sine is -1 at 270°.

7.        A weighted spring oscillates freely between a height of 10 feet below the ceiling to 2 feet below the ceiling.  The amplitude of a representative trigonometric equation would be  equal to 4ft.  Take 10 and subtract 2 and then divide by 2.

 

8.        If  and , find .  Draw your triangle in the 3rd quadrant . . . complete with Pythagorean Thm . . . the tan = 4/3.

 

9.        Simplify completely: .   Remember to change everything to terms of sines and cosines . . . this simplifies to 1.  Everything cancels!! J

 

10.  Find an expression equivalent to    Change sec to 1/cos . . . change tan to sin/cos . . . The sine on the denominator is the same as multiplying by 1/sin.   All of this simplifies to 1/cos2    . . .  .which is equal to sec2.

11.  Find the value of  if .    Draw a triangle for the sine information (and complete Pythagorean) . . . do the same for the cosine information.   Then use your formula for sum of two angles:

Sin(x + y) = sinxcosy + cosxsiny

            Answer:  56/65

12.    Solve .   Solve for cosine.  You’ll get cosΘ = ½ .  Now Θ equals 60° and 300°. 

13.    Solve  for principal values of x.  Solve for x . . . you’ll have to factor out a sinx.  You’ll have 2 answers . . . sinx = 0   OR    sinx = ½    .   So x can equal (in the 1st and 4th quadrants only) 0° or 30°.                           

14.  Given .  Find .      <3, 18>

15.  Write  as the sum of unit vectors for .   Subtract “N” – “M” first . . . .<3, 2, -8> . . . .then as sum of unit vectors . . . 3i + 2j – 8k

16.  Find the magnitude of the vector represented by the vector from .   <-3, -9, -4> . . . . then take the square root of the sum of the squares . . . 9+81+16 =

17.  A pilot flies a plane east for 200 kilometers, then  south of east for 80 kilometers.  Find   

the plane’s distance from the starting point.   This is a law of cosines problem. . .

18.  Given , find a so that the two vectors are perpendicular.    

            -3*4 + 2*a must equal zero . . . . so a = 6.

19.  Find the cross product of , then determine whether the resulting vector is perpendicular to .  Set up a 3x3 matrix . . . use i, j, and k as the first row.  Remember that when you get your answer . . . check to see if a is perpendicular to your answer and if  b is perpendicular to your answer.

20.  Which ordered triple represents  from  to ?   <3, -8, -5>

 

21.   Write an equation of the line in slope-intercept form whose parametric equations are

 

Set each equation equal to t . . . .then cross multiply and solve.  y = -2x - 1

22.  Simplify     FOIL   remember that i2 = -1.    Final answer:  4 -2i

23.   Find the polar coordinates of the point with rectangular coordinates .   (2, 30°)

24.   Find the rectangular coordinates of the point with polar coordinates   (-3,0)

25.  Which pair of polar coordinates also represents the point ?  

  2.    Correct answer!

26.  Rewrite the rectangular equation x = 5 in polar form.   r = 5secΘ

 

 

 

**The first 13 problems on the final will be worked without a calculator.  Below are a few samples of what type of problems to expect:

 

27.  Find sinΘ if Θ is an angle in standard position and the point with coordinates (5,-12) lies on the terminal side of the angle.    If you plot the point (5,-12) . . . the radius is 13.  So, the       sin Θ = -12/13

28.  Solve the following equation for Principal Values:    cos Θ = -1/2  Principal values are in the 1st and 2nd quadrants only for cosine. . . so Θ = 120° and 240°.

29.  On the unit circle, sin (-180) =   0

30.  Which quadrant contains the terminal side if Θ = 2Π/3?   This is 2nd quadrant.

31.  If cos Θ = ¼, find sec Θ.   Secant is the reciprocal of cosine . . . so secΘ = 4.

32.  If Θ is in the 3rd quadrant, and sin Θ = -4/5, find cos Θ.   Draw the triangle and complete the Pythagorean Theorem.  cosΘ = -3/5.

33.  Simplify the following:

                        tan2(Θ)csc2(Θ) =    1/cos2Θ . . . or sec2Θ.

 

 

 

Get rest tonight . . . Bring this review with you tomorrow . . . don’t forget your books!!!

 

See you tomorrow!!!