1) y
=
10)
Graphing Sine and Cosine Functions
For 1 - 3, identify the period and the amplitude of each given function. Simplify if possible.
5 x cos
2 4 y
= −
2
3 sin
π x
10
3) y
=
10
−
3 sin 2 .
5 t
4)
7) y
=
4 cos y
= −
10 x cos
π
6 x
Graph exactly 2 cycles (one on each side of the y -axis) for each function in 4 - 9.
8) y
=
= cos 2
π x
6)
9) y y
=
= sin
−
π sin
4 x
2
π
3 x
State an equation for each graph in 10 - 12.
11) 12)
Pre-Calculus Pre-AP -- Deriving the Sum and Difference Identities
1. Label the coordinates of points A and B in terms of the angles
α
and
β
.
2. Use the Distance Formula to write an expression for the square of the distance from A to B in the picture that we drew in class.
3. Set this expression equal to the one that we got by using the Law of Cosines. Solve this equation for cos
( α − β )
. You will need to square both binomials and simplify.
4. Check that your identity works for
α =
π
2
and
β =
π
3
. When you are certain that your identity is correct, record it in your notes for future use.
5. Next we want to derive a similar identity for cos
( α + β )
= cos
[
α −
] cos
( α + β )
. To do this, observe that
. Substitute into your identity from part 3) and simplify by using some other identities that we have proved previously. Your answer should be in terms of
β
and NOT in terms of
−β
.
6. You can verify that this identity works by letting
α =
π
6
and
β =
π
3
. When you are certain your identity is correct, record it in your notes for future use.
7. Justify the following sequence of statements: sin
( α + β )
= cos
⎡
⎢⎣
π
2
−
( α + β ) ⎤
⎥⎦
= cos
π
2
− α − β = cos
⎡
⎢
π
2
α − β
⎤
⎥ .
8. Use the result of step 7 above and previously derived identities to find an identity for
π answer should NOT contain
2 sin
( α + β )
. Your
anywhere. Check your identity using the same values as in part 6) above and record your new identity in your notes for future use.
9. Use a process similar to steps 7 and 8 above to find an identity for sin
( α − β )
. Check using the values from part 4). Record your new identity in your notes for future use.
10. Use 2 of the identities you just derived to prove that tan
( α + β )
= tan
1
−
α + tan
α tan tan
β
β
. (NOTE: You MUST begin with the LEFT side -- NOT the right side -- on this and on #11 below.)
11. Use the other 2 identities you just derived to prove that tan
( α − β )
= tan
1
+
α − tan
α tan tan
β
β
.