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# 4.4 analytic trigonometry and trig formulas

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## 4.4 analytic trigonometry and trig formulasPresentation Transcript

• Analytic Trigonometry
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
csc() = 1/sin(), read as "cosecant of ".
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
csc() = 1/sin(), read as "cosecant of ".
sec() = 1/cos(), read as "secant of ".
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
csc() = 1/sin(), read as "cosecant of ".
sec() = 1/cos(), read as "secant of ".
cot() = 1/tan(), read as "cotangent of ".
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
csc() = 1/sin(), read as "cosecant of ".
sec() = 1/cos(), read as "secant of ".
cot() = 1/tan(), read as "cotangent of ".
Let (x, y) be a point on the unit circle,then
x=cos() and y=sin()
as shown.
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
csc() = 1/sin(), read as "cosecant of ".
sec() = 1/cos(), read as "secant of ".
cot() = 1/tan(), read as "cotangent of ".
Let (x, y) be a point on the unit circle,then
x=cos() and y=sin()
as shown.
(cos(),sin())
y=sin()
(1,0)

x=cos()
• Analytic Trigonometry
We define the following reciprocals functions that are used often in science and engineering.
csc() = 1/sin(), read as "cosecant of ".
sec() = 1/cos(), read as "secant of ".
cot() = 1/tan(), read as "cotangent of ".
Let (x, y) be a point on the unit circle,then
x=cos() and y=sin()
as shown.
By Pythagorean Theorem,
y2 + x2 = 1, so we have:
sin2() + cos2()=1
or s2 + c2 = 1
or all angle .
(cos(),sin())
y=sin()
(1,0)

x=cos()
• Analytic Trigonometry
The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles .
• Analytic Trigonometry
The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities.
• Analytic Trigonometry
The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities.
The Trig-Hexagram
• Analytic Trigonometry
The equation s2()+ c2() = 1 is called a trig-identity because its true for all angles . The following hexgam contains all the so called fundemantal trig-identities.
The Trig-Hexagram
the CO-side
the regular-side
• Analytic Trigonometry
The Division Relations:
• Analytic Trigonometry
The Division Relations:
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
• Analytic Trigonometry
The Division Relations:
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
II
III
I
• Analytic Trigonometry
The Division Relations:
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
III
I
II
• Analytic Trigonometry
The Division Relations:
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
I
II
III
• Analytic Trigonometry
The Division Relations
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
The Reciprocal Relations
• Analytic Trigonometry
The Division Relations
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
The Reciprocal Relations
Start from any function, going across diagonally, we always have I = II / III.
• Analytic Trigonometry
The Division Relations
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
III
II
I
The Reciprocal Relations
Start from any function, going across diagonally, we always have I = II / III.
Example B: sec(A) = 1/cos(A),
• Analytic Trigonometry
The Division Relations
Start from any function,
going around the outside,
we always have
I = II / III or I * III = II
Example A:
tan(A) = sin(A)/cos(A),
sec(A) = csc(A)/cot(A),
sin(A)cot(A) = cos(A)
I
II
III
The Reciprocal Relations
Start from any function, going across diagonally, we always have I = II / III.
Example B: sec(A) = 1/cos(A), cot(A) = 1/tan(A)
• Analytic Trigonometry
Square-Sum Relations
• Analytic Trigonometry
Square-Sum Relations
For each of the three inverted
triangles, the sum of the squares of the top two is the square of the bottom one.
• Analytic Trigonometry
Square-Sum Relations
For each of the three inverted
triangles, the sum of the squares of the top two is the square of the bottom one.
sin2(A) + cos2(A)=1
• Analytic Trigonometry
Square-Sum Relations
For each of the three inverted
triangles, the sum of the squares of the top two is the square of the bottom one.
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
• Analytic Trigonometry
Square-Sum Relations
For each of the three inverted
triangles, the sum of the squares of the top two is the square of the bottom one.
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
• Analytic Trigonometry
Square-Sum Relations
For each of the three inverted
triangles, the sum of the squares of the top two is the square of the bottom one.
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
The identities from this hexagram are called the
Fundamental Identities.
• Analytic Trigonometry
Square-Sum Relations
For each of the three inverted
triangles, the sum of the squares of the top two is the square of the bottom one.
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
The identities from this hexagram are called the
Fundamental Identities.
Weassume these identities from here on and list the most important ones below.
• Fundamental Identities
Division Relations
tan(A)=S/C
cot(A)=C/S
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square-Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square-Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
B
A
A and B are complementary
Two angles are complementary if their sum is 90o.
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square-Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
B
A
A and B are complementary
Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles,
we have the following co-relations.
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square-Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
B
A
A and B are complementary
Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles,
we have the following co-relations.
sin(A) = cos(90 – A)
sin(A) = cos(B)
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square-Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
B
A
A and B are complementary
Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles,
w have the following co-relations.
sin(A) = cos(90 – A)
tan(A) = cot(90 – A)
sin(A) = cos(B)
tan(A) = cot(B)
• Fundamental Identities
Reciprocal Relations
sec(A)=1/C
csc(A)=1/S
cot(A)=1/T
Division Relations
tan(A)=S/C
cot(A)=C/S
Square-Sum Relations
sin2(A) + cos2(A)=1
tan2(A) + 1 = sec2(A)
1 + cot2(A) = csc2(A)
B
A
A and B are complementary
Two angles are complementary if their sum is 90o.Let A and B = 90 – A be complementary angles,
w have the following co-relations.
sin(A) = cos(90 – A)
tan(A) = cot(90 – A)
sec(A) = csc(90 – A)
sin(A) = cos(B)
tan(A) = cot(B)
sec(A) = csc(B)
• Fundamental Identities
The Negative Angle Relations
cos(-A) = cos(A),
sin(-A) = - sin(A)
A
A
-A
-A
1
1
• Fundamental Identities
The Negative Angle Relations
cos(-A) = cos(A),
sin(-A) = - sin(A)
A
A
-A
-A
1
1
Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).
• Fundamental Identities
The Negative Angle Relations
cos(-A) = cos(A),
sin(-A) = - sin(A)
A
A
-A
-A
1
1
Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).
Therefore cosine is an even function and sine is an odd function.
• Fundamental Identities
The Negative Angle Relations
cos(-A) = cos(A),
sin(-A) = - sin(A)
A
A
-A
-A
1
1
Recall that f(x) is even if anf only if f(x) = f(-x), and that f(x) is odd if and only iff f(-x) = -f(x).
Therefore cosine is an even function and sine is an odd function.
Since tangent and cotangent are quotients of sine
and cosine, they are also odd functions.
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
cot2(A) – csc2(A) = -1
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
cot2(A) – csc2(A) = -1
Difference of squares may be factored since
x2 – y2 = (x – y)(x + y)
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
cot2(A) – csc2(A) = -1
Difference of squares may be factored since
x2 – y2 = (x – y)(x + y)
Example D: (1 – sin(A))(1 + sin(A))
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
cot2(A) – csc2(A) = -1
Difference of squares may be factored since
x2 – y2 = (x – y)(x + y)
Example D: (1 – sin(A))(1 + sin(A))
= 1 – sin2(A)
 Frank Ma
2006
• Notes on Square-Sum Identities
Terms in the square-sum identities may be rearranged and give other versions of the identities.
Example C: sin2(A) – 1 = – cos2(A)
sec2(A) – tan2(A) = 1
cot2(A) – csc2(A) = -1
Difference of squares may be factored since
x2 – y2 = (x – y)(x + y)
Example D: (1 – sin(A))(1 + sin(A))
= 1 – sin2(A) = cos2(A)
 Frank Ma
2006
• Algebraic Identities
Example E: Simplify (sec(x) – 1)(sec(x) + 1).
Express the answer in sine and cosine.
• Algebraic Identities
Example E: Simplify (sec(x) – 1)(sec(x) + 1).
Express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1)
• Algebraic Identities
Example E: Simplify (sec(x) – 1)(sec(x) + 1).
Express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) ;difference of squares
= sec2(x) – 1
• Algebraic Identities
Example E: Simplify (sec(x) – 1)(sec(x) + 1).
Express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) ;difference of squares
= sec2(x) – 1 ;square-relation
= tan2(x)
• Algebraic Identities
Example E: Simplify (sec(x) – 1)(sec(x) + 1).
Express the answer in sine and cosine.
(sec(x) – 1)(sec(x) + 1) ;difference of squares
= sec2(x) – 1 ;square-relation
= tan2(x) ; division relation
=
sin2(x)
cos2(x)