UNIT 5 – RIDING ON THE FERRIS WHEEL LEARNING TASK:
Standard: F.TF.9 Prove addition, subtraction, double, and half-angle formulas for sine, cosine, and tangent and use them to
solve problems.
Essential Question: What is a double angle identity? Does doubling an angle in a trig function, double the output of the trig
function? What is a half angle identity?
Use this diagram to answer
Questions #1 through #5
1.
Lucy is riding a Ferris wheel with a radius
of 40 feet. The center of the wheel is
55 feet off of the ground, the wheel is
turning counterclockwise, and Lucy is
halfway up the Ferris wheel, on her
way up. Label the picture above with
Lucy’s position and any other
measurements you know.
2.
If the wheel makes a complete turn every
1.5 minutes, through what angle, in degrees,
does the wheel turn each second?
3.
Label Lucy’s position 10 seconds after passing her initial position in problem 1? What height is Lucy at in this picture?
4.
Label Lucy’s position 20 seconds after passing her initial position in problem 1? What height is Lucy at in this picture?
5.
Label Lucy’s position 40 seconds after passing her initial position in problem 1? What height is Lucy at in this picture?
6.
Write an expression that gives Lucy’s height t seconds after passing her position in problem 1, in terms of t.
7.
In problems 4 and 5, the angle through which Lucy turned was twice that of the problem before it. Complete the
table below.
Lucy’s Turn Angle
Lucy’s Height
Did her change in height double as well?
Students commonly think that if an angle doubles, then the sine of the angle will double as well, but as you saw in the
previous problems, this is not the case. The double angle identities for sine and cosine describe exactly what happens to these
functions as the angle doubles. These identities can be derived directly from the angle sum identities, printed here for your
convenience:
cos      cos  cos   sin  sin 
8.
sin      sin  cos   cos sin 
Derive the double angle identity for cosine by applying the angle sum identity.
cos 2   cos     
Could this formula could be rewritten as cos(2θ) = 2 cos2θ – 1 or 1 – 2 sin2θ. How?
9.
Derive the double angle identity for sine, also by applying the angle sum identity.
sin 2   sin     
10. Derive the double angle using the identity for tangent as being sin 2𝜃/ cos 2𝜃.
tan 2𝜃 =
After deriving the double-angle identities for sine, cosine, and tangent, you’re ready to try the same for half-angle identities.
To do so, you will use the double-angle identities you just derived.
11.
Begin with either of the alternate forms of the cosine double-angle identity. (Notice that, in the alternative identities,
there are two instances of the variable x – one that is x alone and the other that is 2x.)
12.
Rearrange your chosen identity so that the term with x alone gets isolated on a side. (Think: If you wanted to evaluate
angle x for the trig function you isolated, what information, would you need to know in order to use the identity you
have?)
13.
Now we have an identity that will give us the trig function value of half an angle if we know a trig function value of the
𝑢
full angle. We can do this with a simple substitution – substitute = 𝑥 and simplify. Record your results below.
2
14.
𝑢
Now derive the tan identity using the quotient identity for tangent.
2
Assignment #1: Use the half-angle identities to find the exact value for sin, cos and tan of the following:
𝜋
𝜋
1.
15◦
2.
165◦
3.
4.
8
15.
12
Given sin θ = 3/5, find the following:
a. cos θ
b.
sin 2θ
16.
Given sin β = .7, find sin 2β and cos 2β
17.
Given cos α = .6, find sin 2α
18.
Given cos u =
19.
Given cot u = -6 and
20.
Given sin 𝑢 =
−2
7
and
5
13
𝜋
2
3𝜋
2
and
c.
< 𝑢 < 𝜋, find sin(2u), cos(2u) and tan(2u)
< 𝑢 < 2𝜋, find sin(2u), cos(2u) and tan(2u)
𝜋
2
𝑢
𝑢
2
2
< 𝑢 < 𝜋, find sin , cos
and tan
𝑢
2
cos 2θ
5.
d.
3𝜋
8
sin 4θ
Assignment #2 – Complete on your own paper
1.
Given cos β = 2/3, find sin 2β and cos 2𝛽
b
2.
Given tan β = ½, find sin 2β and cos 2β
3.
Given the triangle on the right, find the following:
a.
sin a =
b.
cos a =
5
a
4.
c.
tan a =
d.
sin 2a =
e.
cos 2a =
f.
sin b =
g.
cos b =
h.
tan b =
i.
sin 2b =
j.
cos 2b =
Given cos 𝑢 =
a.
5.
3.
7
8.
sin
𝑢
7
12
𝜋
and 0 < 𝑢 < , find the following:
25
2
b.
2
cos
𝑢
c.
2
5
tan
𝑢
2
3
Find the exact value of the trig function given that sin u = and cos v = - . Both u and v are in Quadrant II. (Hint:
13
5
Draw the triangles first and identify the other needed functions. Pay attention to Quadrant II signs.)
a.
sin (u + v)
b.
cos (u - v)
c.
cos (u + v)
d.
sin (v - u)
e.
tan (u + v)
g.
sec (v - u)
h.
cot (u + v)
Find the exact value of the trig function given that sin u =
−7
25
and cos v =
f.
−4
5
csc (u - v)
. Both u and v are in Quadrant III.
a.
cos (u + v)
b.
sin (u + v)
c.
tan (u - v)
d.
cot (v - u )
e.
sec (u + v)
f.
cos (u - v)
4
3𝜋
5
2
Given sin 𝐴 = − , 𝜋 < 𝐴 <
, sec B =
13
5
𝜋
and 0 < 𝐵 < , find each of the following:
2
a.
sin (A + B)
b.
cos (A + B)
c.
sin (A – B)
d.
cos (A – B)
e.
sin(2B)
f.
cos(2A)
g.
sin ( )
h.
cos ( )
𝐴
2
4
5
5
13
Given sin A = , A is in Quadrant I, cos B = -
𝐴
2
, B is in Quadrant II, find each of the following:
a.
sin (A + B)
b.
cos (A + B)
c.
sin (A – B)
d.
cos (A – B)
e.
tan (A + B)
f.
csc (A – B)
Assighment #3
Practice:
Given cos α = 
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
1.
First draw the triangle in quadrant 2.
Label the two sides of the triangle you already know.
Use Pythagorean theorem to find the missing side.
Find sin α
Find sin 2α (Remember sin 2α = 2 sin α cos α)
Find cos 2α (remember cos 2α = cos2 α – sin2 α)
If tan α =
a.
b.
c.
d.
2.
3
5
cos α
sin 2α
cos 2α
Given sin α = 
12
and α is in the 4th quadrant, find the following:
13
cos α
sin 2α
cos 2α
Given tan α =
a.
b.
c.
d.
6.
cos α
sin 2α
cos 2α
Given sin α =  and α is in the 4th quadrant, find the following:
a.
b.
c.
5.
sin α
cos α
sin 2α
cos 2α
3
5
a.
b.
c.
4.
5
and α is in the 3rd quadrant, find the exact values of each of the following.
12
Given sin α =  and α is in the 3rd quadrant, find the exact values of each of the following.
a.
b.
c.
3.
5
and α is in the 2nd Quadrant, determine the exact values of: (a) sin 2α and (b) cos 2α
13
12
and α is in the 1st quadrant, find the following:
5
sin α
cos α
sin 2α
cos 2α
Given cos α = 
a.
b.
c.
sin α
sin 2α
cos 2α
7
and α is in the 2nd quadrant, find the following:
25