t06 ⊾ 08 July 2011 ⊿ Name:

1. ___________________ Simplify 2 sin(θ) cos(θ) Write only the first result.
2. ___________________ Simplify (1 − cos²θ)(1 − cot²θ) Write only the first result.
3. _________ The equation for a projectile is given by: $\text{distance}=\frac{2v^2\sin \left(\text{θ}\right)\cos \left(\text{θ}\right)}{\mathrm{9.8}}$
   For a velocity v of 14 m/s, and an angle of 30°, calculate the distance d.
4. _________ Solve for the velocity v: $\frac{2v^2\sin \left(30\right)\cos \left(30\right)}{\mathrm{9.8}}=\mathrm{80}$ with the angles given in degrees.
5. _________ Solve for the angle x: 19.6 sin(x) cos(x) = 9.8
   Report the angle x in radians for the solution between 0 and π/2.
6. (_______, _______) For my physical science test I needed to move an electron at position a to position b. Note that in this example (0, 0) is at the center of the circle. Position a is (170,0). See the diagram on the below left. What are the coordinates of electron b?
   
7. ____________ Calculate the magnitude (vector length) for the vector {80, 80}. Note that in the diagram on the above right all vectors are in the standard position with their initial point at the origin (0, 0).
8. ____________ Calculate the angle relative to the horizontal axis for the vector {80, 80}.
9. ____________ Calculate the magnitude for the vector {−86.61, 50}.
10. ____________ Calculate the angle relative to the horizontal axis for the vector {−86.61, 50}.
11. ____________ Calculate the magnitude for the vector {−40, −30}.
12. ____________ Calculate the angle relative to the horizontal axis for the vector {−40, −30}.
13. ____________ Calculate the magnitude for the vector {95, 0}.
14. ____________ Calculate the angle relative to the horizontal axis for the vector {95, 0}.
15. 
    On a Thursday evening in July 2005 I was body board surfing in Malem. My total velocity v (speed) was the result of the vector addition of the velocity (speed) of the wave w and the velocity s of my board along the face of the wave. On that Thursday the waves were moving at a speed of:
    $\stackrel{\rightharpoonup }{w}=3.9$ m/s due East towards Malem and my board was moving at
    $\stackrel{\rightharpoonup }{s}=5.2$ m/s due North towards Lelu.
    Note that w and s are at right angles as shown in the inset diagram.
16. Calculate my velocity v over the reef: __________
17. _________ Calculate the angle between v and w.

Children body boarding in the surf off of Piyuul, Malem.