Precalculus
Take Home Test 2
Each problem with be scored on the 6 point rubric I described and an additional 6 points awarded for neatness, clarity of
solutions, and overall presentation of ideas. Please do all work on separate paper and do not include this sheet with your work.
Your responses should stand on their own! I expect high quality solutions since you have so long to work on these.
Use the Plan below the diagram and:
1.
Derive the identity for sin(    ) geometrically
2.
Derive the identity for cos(   ) geometrically
You are expected to
provide your own
illustration for the final
write-up of your solutions!
Plan for #1 and #2
1. Label the coordinates in terms of cos  , sin  , cos(   ) , and sin(    ) .
2. Make a right triangle by drawing the side opposite  to terminal side of  .
3. Also label the sides represented by cos  and sin  .
4. Draw a vertical line segment that represents sin(    ) . Label the other angle  .
5. Label line segments a and b. a goes from the x –axis to the intersection of d and sin(    ) . b is
the remaining distance to the point ( cos(   ) , sin(    ) )
6. Draw and label other segment a to form a rectangle.
7. We are going to concentrate on the "little triangle" at the top with angle  and leg b, and
hypotenuse sin  and the "big triangle" at the bottom with angle  and opposite leg a, with
hypotenuse cos  .
8. Label other segment d.
9. Derive the identities for sin(    ) and cos(   ) geometrically by using the information on
your diagram.
(Use right triangle trig!)
3.
Disprove that sin(    )  sin   sin  by providing a counterexample.
4. The figure shows a vertical cross section through a piece of land. The y-axis is drawn coming out of the ground at the
fence bordering land owned by your boss, Earl Wells. Earl owns the land to the left of the fence and is interested in
buying land on the other side to drill a new oil well. Geologists have found an oil-bearing formation, which they
believe to be sinusoidal in shape, beneath Earl’s land. At x = – 100 feet, the top surface of the formation is at it’s
deepest, y = – 2500 feet. A quarter-cycle closer to the fence, at x = – 65 feet, the top surface is only 2000 feet deep.
The first 700 feet of land beyond the fence is inaccessible. Earl wants to drill at the first convenient site beyond x =
700 ft.
Using y  a cos b( x  c)  d find the sinusoidal equation for y as a function of x.
Plot the graph on your grapher. Use a window with an x-range of [-100 , 900].
Find graphically the first interval of x-values in the available land for which the top surface of the formation is no
more that 1600 feet deep. Then show algebraically the values of x at the end of that interval.