Download in the interval - Fort Thomas Independent Schools

P.o.D.
1.) Write
sin 60° cos 45° − cos 60° sin 45° as a single trig function.
2.) Find all solutions of
2𝑠𝑖𝑛2 𝑥 − 3 sin 𝑥 = −1 on the interval [0,2𝜋].
3.) If sin(x)=(3/5) and cos(x)=(4/5), find tan (x).
5.5 – Multiple Angle and Product-to-Sum Formulas
Learning Target: I will be able to use multiple-angle and half-angle formulas to
evaluate trigonometric functions.
Essential Question: How do you rewrite trigonometric expressions that contain
functions of multiple or half-angles, or functions that involve square or products
of trigonometric expressions?
Double Angle Formulas:
sin 2𝑢 =
cos 2𝑢 =
cos 2𝑢 =
cos 2𝑢 =
tan 2𝑢 =
EX: Solve cos 2𝑥 + cos 𝑥 = 0
Use a double angle identity
Factor by ____________
Set each equation equal to zero.
EX: Use a double-angle formula to rewrite the equation
𝑔(𝑥) = 3 − 6𝑠𝑖𝑛2 𝑥. Then sketch the graph of the equation over the interval
[0,2𝜋].
Begin by factoring.
We should now recognize a double-angle identity for __________.
Graph.
(show a detailed graph on
the whiteboard)
3
𝜋
5
2
EX: Use sin 𝑢 = , 0 < 𝑢 < to find sin 2u, cos 2u, and tan 2u.
We must recognize that this is in Quadrant __.
In order to use the double-angle identity for sine, we need both ______ and
__________. Use a _______________ Identity to find cosine.
Now we can apply the _________________________ identity for sine.
We have three possible identities for cos 2u. We will use ________________ since
___________ was the given value.
We need to find __________ in order to use the double angle identity for tangent.
Now we can use the double-angle identity for tangent
EX: Derive a triple-angle formula for cos(3x).
Begin by writing 3x as a sum of 2 angles.
Now apply the ____ identity for ____________.
Next, apply a double angle identity for 2x.
Simplify.
Rewrite everything in terms of one trig function.
Simplify.
Half-Angle Formulas:
sin
𝑢
=
2
cos
𝑢
=
2
tan
𝑢
=
2
*Remember that the sign must always be consistent with the quadrant.
EX: Find the exact value of cos 105°
We know that our answer must be _____________ since ____ degrees is in ___.
EX: Find the exact value of tan 22.5°
𝑥
EX: Solve 𝑐𝑜𝑠 2 𝑥 = 𝑠𝑖𝑛2 in the interval [0, 2𝜋]
2
Apply the half-angle identity for sine.
Simplify. (cross multiply)
Set up a quadratic.
Solve by factoring.
*Remember, we can always confirm our answers graphically.
Product to Sum Formulas:
sin 𝑢 sin 𝑣 =
cos 𝑢 cos 𝑣 =
sin 𝑢 cos 𝑣 =
cos 𝑢 sin 𝑣 =
EX: Rewrite sin 5𝜃 cos 3𝜃 as a sum or difference.
Sum to Product Formulas:
sin 𝑢 + sin 𝑣 =
sin 𝑢 − sin 𝑣 =
cos 𝑢 + cos 𝑣 =
cos 𝑢 − cos 𝑣 =
EX: Find the exact value of
sin 195° + sin 105°
EX: Solve sin(4x)-sin(2x)=0 on the interval [0,2𝜋]
*Let’s solve this graphically.
EX: Verify the identity
sin 6𝑥+sin 4𝑥
cos 6𝑥+cos 4𝑥
= tan 5𝑥
EX: Ignoring air resistance, the range of a projectile fired at an angle 𝜃 with the
horizontal and with an initial velocity of 𝑣0 feet per second is given by 𝑟 =
1
𝑣
16 0
2
sin 𝜃 cos 𝜃 where r is the horizontal distance (in feet) that the projectile will
travel. A place kicker for a football team can kick a football from ground level with
an initial velocity of 78 feet per second.
a.)
At what angle must the player kick the football so that it travels 188
feet?
b.)
For what angle is the horizontal distance the football travels a
maximum?
Let’s solve this graphically.
The ball will travel a maximum distance when it reaches its maximum range.
Therefore, the best angle is ____ degrees.
HW
Pg.415
6-90 6ths, 99, 119.
Quiz 5.4-5.5 tomorrow