Transcript and cos (A + B)

WARM UP
tan x =
sec x
csc x
Quotient
is caled a ___________
property.
The value of 30 is what percentage of 1000? 3%
cos(7 – 3) = cos 4
cos 7 cos 3 + sin 7 sin 3 = ________
What is the amplitude of the sinusoid y = 8 cos θ + 15
sin θ? 17
If A cos (θ – D) = 8 cos θ + 15 sin θ, then D could equal
15
arctan  61.9275....
8
OTHER COMPOSITE
ARGUMENT
PROPERTIES
OBJECTIVES
For trigonometric functions f, derive and
learn properties for:
 f(-x) in terms of f(x)
 f(90 – θ) in terms of functions of
θ, or f(π/2 – x)
in terms of functions of x
 f(A +B) and f(A – B) in terms of functions of A
and functions of B
KEY WORDS &
CONCEPTS
Odd-even
properties
 Parity
 complementary
 Cofunction properties
 Composite argument
properties
 Triple argument
properties
ODD EVEN PROPERTIES
If you take the function of opposite angles or arcs,
interesting patterns emerge.
sin (-20°) = -0.3420…
and
sin 20° = 0.3420…
cos (-20°) = 0.9396…
and
cos 20° = 0.9396…
tan (-20°) = -0.3639…
and
tan 20° = 0.3639…
DEFINING MEASUREMENT
OF ROTATION
These numerical examples illustrate the fact that sine and tangent
are opposite functions and cosine is an even function. The graphs
show why these properties apply for any value of θ.
The reciprocals of the function have the same parity
(oddness or evenness) as the original functions.
PROPERTIES
ODD AND EVEN FUNCTIONS
Cosine and its reciprocal are even functions. That is,
cos (-x) = cos x
and
sec (-x) = sec x
Sine and tangent, and their reciprocals, are odd functions.
That is,
sin (-x) = -sin x
tan (-x) = -tan x
and
and
csc (-x) = -csc x
cot (-x) = -cot x
COFUNCTIONS PROPERTIES
Properties of (90° - θ) or (π/2 – x)
The angles 20° and 70° are complementary angles because
they sum to 90°. (The word comes from “complete,” because
the two angles complete a right angle.) The angle 20° is the
complement of 70°, and the angle 70° is the complement of
20°. An interesting pattern shows up if you take the function
and the co-function of complementary angles.
cos 70° = 0.3420…
cot 70° = 0.3639…
and
and
sin 20° = 0.3420…
tan 20° = 0.3639…
csc 70° = 1.0641…
1.0641…
and
sec 20° =
PROOF
You can verify these patterns by using the right triangle
definitions of the trigonometric functions. The right
triangle with acute angles measure 70° and 20°. The
opposite leg for 70° is the adjacent leg for 20°. Thus,
adj.leg a
cos70 

hypotenuse c
and
opp.leg
a
sin20 

hypotenuse c
COFUNCTIONS
The prefix co-in the names cosine, cotangent and cosecant
comes from the word complement. In general, the cosine
of an angle is the sine of the complement of that angle.
The same property is true for cotangent and cosecant, as
you can verify in the previous triangle .
The co-function properties are true regardless of the
measure of the angle or arc. For instance, if θ is 234°, the
the complement of θ is 90° - 234°, or = -144°.
cos 234° = −0.5877… and
sin (90° − 234°) = sin (−144°) =
−0.5877…
cos 234° = sin (90° − 234°)
COFUNCTION PROPERTIES
Note that it doesn’t matter which of the two
angles you consider to be “the angle” and which
you consider to be the “complement. It is just as
true, for example, that sin 20° = cos (90° - 20°)
The cofunction properties for trigonometric functions
are summarized verbally as
•The cosine of an angle equals the sine of the
complement of that angle.
•The cotangent of an angle equals the tangent of the
complement of that angle
•The cosecant of an angle equals the secant of the
complement of that angle.
PROPERTIES
Cofunction Properties for Trigonometric Functions
When working with degrees:
cos θ = sin (90° − θ)
and
(90° − θ)
cot θ = tan (90° − θ)
and
− θ)
csc θ = sec (90° − θ)
and
(90° − θ)
When working with radians:
cos x = sin ( − x)
and
− x)
cot x = tan ( − x)
and
sin θ = cos
tan θ = cot (90°
sec θ = csc
sin x = cos (
tan x = cot (
COMPOSITE ARGUMENT PROPERTY
for cos (A + B)
You can write the cosines of a sum of two angles in terms
of functions of those two angles. You can transform the
cosine of a sum to a cosine of a difference with some
insightful algebra and the odd-even properties.
cos (A + B)
Change the sum into a difference.
= cos [A – (-B)]
cos A cos (-B) + sin A sin (B)
= cos A cos B + sin A (-sin
B)
= cos A cos B – sin A sin B
=
Use the composite argument property for
cos (first – second)
Cosine is an even function. Sine is an odd
function
cos (A + B) = cos A cos B – sin A sin
TheBonly difference between this property and the one for cos (A – B) is
=
the sign between the terms on the right side of the equation.
COMPOSITE ARGUMENT PROPERTY
for sin (A – B) and sin (A + B)
You can derive composite argument properties for sin (A –
B) with the help of the cofunction property.
sin (A – B) = cos [90 – [A –
B)]
= cos [(90° – A) + B]
=
Transform into a cosine using the cofunction
property
Distribute the minus sign, then associate
(90° – A)
cos (90 – A )cos B – sin (90° – A) sin B
Use the composite argument property for
cos (first + second)
=
sin A cos B – cos A sin B
The composite argument property for sin (A +
B)
sinis(A + B) = sin A cos B + cos A sin
You B
can derive it by writing sin (A + B) as sin [A – (-B)] and using the
same reasoning as for cos (A + B).
COMPOSITE ARGUMENT PROPERTY
for tan (A – B) and tan (A + B)
You can write the tangent of a composite argument in terms of
tangents of the two angles. This requires factoring out a
“common” factor that isn’t actually there!
sin(A  B)
tan(A  B) 
cos(A  B)

sin A cos B  cos Asin B
cos A cos B  sin Asin B
sin A cos B

cos
A
cos
B

cos A cos B
cos A cos B

cos A cos B
cos A cos B
sin A sin B

cos
A
cos B

sin Asin B
1
cos A cos B

Us the quotient property for tangent to
bring in” sines and cosines.
tan A  tan B
1 tan A
cos Asin B
cos A cos B
sin Asin B
cos A cos B
Use the composite argument properties for
sin (A – B) and cos (a – B).
Factor out (cos A cos B) in the numerator
and
denominator to put cosines in the minor
denominator.
Cancel all common factors.
Use the quotient property to get only tangents.
SOLUTIONS
You can derive it by writing sin (A + B) as sin [A – (-B)]
and using the same reasoning as for cos (A + B).
tan A  tan B
tan(A  B) 
1  tan A tan B
The box on the next page summarizes the composite
argument properties for cosine, sine and tangent. As with the
composite argument properties for cos (A – B) and cos (A + B),
notice the signs between the terms change when you compare
sin (A – B) and sin (A + B) or tan (A – B) andtan (A + B).
PROPERTIES
Composite Argument Properties for Cosine, Sine, and Tangent
cos (A – B) = cos A cos B + sin A sin B
cos (A + B) = cos A cos B – sin A sin B
sin (A – B) = sin A cos B – cos A sin B
sin (A + B) = sin A cos B + cos A sin B
tan A  tan B
tan(A  B) 
1 tan A tan B
tan A  tan B
tan(A  B) 
1  tan A tan B
ALGEBRAIC SOLUTION OF
EQUATIONS
You can use the composite argument properties to
solve certain trigonometric equations algebraically.
Example 1: Solve the equation for x
solutions graphically.
Solution:
 [0, 2π].
1
sin 5x cos 3x  cos 5x sin 3x 
2
1
sin 5x cos 3x  cos 5x sin 3x 
2
1
sin(5x  3x) 
2
1
sin 2x  arcsin
2
Verify the
Write the given equation.
Use the composite argument property for sin (A
– B)
SOLUTION CONTINUED
2x = arcsine 1/2
= 0.5235… + 2πn
or
(π – 0.5235…) + 2πn
Use the definition of arcsine to write the
general solution.
x = 0.2617… + πn
or
1.3089… + πn
x = 0.261…, 1.3089…, 3.4033…, 4.4505…
Use n = 0 and n = 1 to get the solutions in the
domain.
In this case, the answers turn out to be simple multiples of
π. See if you can figure out why x =  5 13 
17
,
,
,
12 12 12 12
,and
12
.
SOLUTION CONTINUED
The graph show y = sin 5x cos 3x – cos 5x sin 3 and the line
y = 0.5. Note that the grpha of y = sin 5x cos 3x – cos 5x
sine 3x is equivalent to the sinusoid y – isn 2x. By using
the intersect feature, you can see that th four solutions
are correct and that they are the only solutions in the
domain

x
[0, 2π]
CH. 5. 3 HOMEWORK
Textbook pg. 211 #2-34 every other even