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4.4 analytic trigonometry and trig formulas
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    4.4 analytic trigonometry and trig formulas 4.4 analytic trigonometry and trig formulas Presentation 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)