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Mr. Geocaris Honors Algebra II
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Double Angle Theorem and Identites
Even and Odd Functions
Formulas to Know
Half Angles
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Practice Identities
Proving Trigonometric Identities
Simplifying Trigonometric Expressions
Solving Trigonometric Equations
Sum and Product
SumDifference Identities
The Unit Circle
Trigonometric Identities
Trigonometry In Real Life
SumDifference Identities
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difference
double angle
identities
proof
radians
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tangent
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Sum, Differ____ence, & Double Angle Identities
Unit CircleUNIT CIRCLE
Proof Practice
Angle Difference Formula
NOW YOU TRY IT! Try these sample problems. Just highlight the text to see the answer!
Angle Sum Formula
SUMDIFFERENCE IDENTITIES:
Sum, Differ
ence, & Double Angle Identities
Angle Sum and Difference Theorem:
The following identities are true for all values for which they are defined:
♦
cos(x  y) = cos(x)cos(y) + sin(x)sin(y)
cos(x + y)=cos(x)cos(y)  sin(x)sin(y)
sin(x  y)=sin(x)cos(y)  cos(x)sin(y)
sin(x + y)=sin(x)cos(y) + cos(x)sin(y)
gsdfdgdfg
tan(x)+tan(y)
tan(x+y)= 
as asdf
1tan(x)tan(y)
sdfggsdfg
tan(x)tan(y)
tan
(xy)= 
sssssss
1+tan(x)tan(y)
Although the equations look long, it is really just simply a matter of plugging in the numbers.
Don't get too worked up about the x's and y's, you could even substitute them for A and B if you'd like.
As long as x sine and x cosine are from the same number and the y sine and y cosine are from the
same, you should be good to go.
Here is an example of proving sum and difference identities.
cos(xy) + cos(x+y) = 2cos(x)cos(y)
cos(x)cos(y) + sin(x)sin(y) + cos(x)cos(y)  sin(x)sin(y) = 2cos(x)cos(y)
2cos(x)cos(y) = 2cos(x)cos(y)
>> Here is a "cheat sheet" for when you start using Tangent from the Unit Circle in the formulas state above.
Don't forget!!! To find tangent, the formula is tan=
y/x, (meaning tangent =
y over x).
Degrees
Radians
Tangent Measure
30°
π
/6
(√3)/3
45°
π
/4
1
60°
π
/3
(√3)
90°
π
/2
Undefined
120°
2
π
/3
(√3)
135°
3
π
/4
1
150°
5
π
/6
(√3)/3
180°
π
0
210°
7
π
/6
(√3)/3
225°
5
π
/4
1
240°
4
π
/3
(√3)
270°
3
π
/2
Undefined
300°
5
π
/3
(√3)
315°
7
π
/4
1
330°
11
π
/6
(√3)/3
360°
2
π
0
Using sum and difference identities
Unit Circle
UNIT CIRCLE
Proof Practice
Proof:
Let
P
be the point with coordinates
. Measured counterclockwise from point
P
, let
Q
be the point whose arc length is
A
, let
R
be the point whose arc length is
, and let S be the point whose arc length is
. Then
are the coordinates of point
Q
,
are the coordinates of point
R
, and
are the coordinates of point
S
.
Note that the lengths of segments
PR
and
QS
are equal. Using the distance formula, we get:
Through the use of the
symmetric
and
Trigonometric Identities
, this simplifies to become the angle sum formula for the cosine.
The formulas can also be derived using triangles. Although we refer to the following derivation as a proof, in fact the values of angles
A
and
B
allowed by the derivation are quite limited, and a more general proof is actually required
.
Angle Difference Formula
sin(A − B) = sinAcosB − cosAsinB
cos(A − B) = cosAcosB + sinAsinB
These formula allow you to express the exact value of trigonometric expressions that you could not otherwise express. Consider the cos(15°). Unlike the cos(60°) which can be expressed as ½, the cos(15°) cannot simply be represented as a rational expression. However, the angle difference formula allows you to represent the exact value of this function.
Picture of angle difference formula
NOW YOU TRY IT! Try these sample problems. Just highlight the text to see the answer!
cos15=?
cos15=cos(6045)
We know that cos(xy)=cosxcosy+sinxsiny, so
cos(6045)=cos60cos45+sin60sin45
cos(6045)=(1/2)(√2/2)+(√3/2)(√2/2)
cos(6045)=((√2)+√6)/2
Angle Sum Formula
sin(A+B) = sinAcosB + cosAsinB
cos(A+B) = cosAcosB − sinAsinB
These formulas allow you to express the exact value of trigonometric expressions that you could not otherwise express. Consider the sin(105°). Unlike the sin(30) which can be expressed as ½, the sin(105) cannot simply be represented as a rational expression. However, the angle sum formula allows you to represent the exact value of this function
Picture of angle sum formula
So to wrap things up:
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