IMSA
MI-4
Prob. Set #10
DUE: Friday, November 30
Fall 2012
cos 25  sin 25 cos15  sin15
1) NC Let A = 


 sin 25 cos 25   sin15 cos15 
a. Find and simplify the matrix A using trigonometry and the properties of matrices. (Your
answer should be exact and may involve terms such as cos 40,sin10 , or similar expressions).
b. Explain why this result should have been expected in light of your previous unit and
transformations.
 5 1 
2) Simplify    
i 0  i   3 
from before.)
5
5i
i
2
  (Looks like something to do with the Naomi and Shawon problem
3
3) NC Find the exact value of each sum:

a.

k 0
4 – 2k  1
4k
b.
24  20 
50 125


3
9

 1 
4) NC Find the exact value of sin  2sin 1    .
 4 

5) Solve the following for x, 0  x < 2:
sin x
1
1 + cos x
6) Asher loves to play on the swings at the park. He jumps from the swing when it is at the top of its arc
(60 inches above the ground)1. If his horizontal velocity is 150 inches per second, then his position at
any time t is given by:
v  x(t )iˆ  y (t ) ˆj , where x(t )  150t  150 and y (t )  60  192t 2
We measure t in seconds (after the jump). The x and y are measured in inches from the point, O, on the
ground directly under the resting swing. How far from point O does Asher land?
7) NC Given that sin x  cos x 
1
8
, find sin x  cos x
25
Do not attempt this without trained professionals present.
IMSA
MI-4
Prob. Set #10
DUE: Friday, November 30
Fall 2012
8) NC For what value(s) of a are the vectors, aiˆ  (a  1) ˆj  (a  3)kˆ and (a  1)iˆ  (a  2) ˆj  akˆ
orthogonal?
9)
Find, to the nearest minute, the angle formed by two diagonals of a cube.
10)
In ∆ABC, a + sinA = b + sinB. What can you tell about this triangle?
x  t
11) The parametric form of the rectangular equation y  x 2 is 
. You can convince yourself of
2
y  t
this by removing the parameter t. Use a computer to graph the parametric form of this parabola and
attach it to your answer sheets.
General Rotation Formula for Parametric Equations
The rotation of any parametric equations through an angle  is given by
 x  x 'cos  y 'sin 

 y  x 'sin   y 'cos
where x ' is the original equation for x and y ' the original equation for y.
 x '  cos
These formulas came from multiplying the rotation matrix R     
 y '  sin 
 sin    x ' 
cos   y '

of the parametric equations you graphed in
4
problem #11. Plot this on the same grid as problem #11.
12) Find the parametric equations of the rotation of
13) There is a theorem in mathematics that states any conic section can be written
Ax 2  Bxy  Cy 2  Dx  Ey  F  0 .
In the case of y  x 2 , A  1 , E  1 , and all other coefficients are zero.
Find the rectangular equation of the 45º rotation of the parabola you graphed in problem #12.
Note: Calculators OK but show enough work to follow your reasoning.
IMSA
MI-4
Prob. Set #10
DUE: Friday, November 30
Fall 2012