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Chapter 6

Chapter 6. ADDITIONAL TOPICS IN TRIGONOMETRY. 6.1 Law of Sines. Objectives Use the Law of Sines to solve oblique triangles Use the Law of Sines to solve, is possible, the triangle or triangles in the ambiguous case Find the area of an oblique triangle using the sine function

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Chapter 6

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  1. Chapter 6 ADDITIONAL TOPICS IN TRIGONOMETRY

  2. 6.1 Law of Sines • Objectives • Use the Law of Sines to solve oblique triangles • Use the Law of Sines to solve, is possible, the triangle or triangles in the ambiguous case • Find the area of an oblique triangle using the sine function • Solve applied problems using the Law of Sines

  3. Law of Sines • Previously, our relationships between sides of a triangle and the angles were unique only to RIGHT triangles • What about other triangles? Obtuse or acute ones? (oblique – not right!) • The following relationship exists (A,B,C are measures of the 3 angles; a,b,c are the lengths of sides opposite those angles):

  4. Solving an oblique triangle • If given: A = 50 degrees, B = 30 degrees, b = 7 cm. Can you solve this triangle? If so, is the solution unique? • You know C = 100 degrees (Now that you know the measure of all 3 angles and the length of 1 side, how many triangles exist? Only one!) • You can find “a” & “c” by law of sines:

  5. What if given: a=5”,b=7”,B=45 degrees • Law of Sines indicates sinA=.505. There are 2 angles,A, such that sin(A) = .505. A = 30 degrees, OR A = 150 degrees (WHY? Look at your unit circle!) • Are there 2 possible triangles? NO – in this case, a (5”)is smaller than b (7”), and if angle A = 150 degrees, it must be opposite the longest side of the triangle. Clearly, it is not, therefore only 1 triangle exists. (continued)

  6. Example continued • If A = 30 degrees & B = 45 degrees, C=105 degrees • Use law of sines again to find c.

  7. What if given: a=7”,b=5”,B=45 degrees • Law of Sines indicates sinA=.99. There are 2 angles,A, such that sin(A) = .99. A = 82 degrees, OR A = 98 degrees (WHY? Look at your unit circle!) • Are there 2 possible triangles? YES – in this case, a (7”)is larger than b (5”), and if angle A = 82 or 98 degrees, it is a larger angle than B. Clearly, there are 2 triangles that exist. (continued)

  8. 2 possible triangles • Triangle 1: use law of sines to find c: • Triangle 2: use law of sines to find c:

  9. Finding area of an oblique triangle • Using Law of Sines it can be found that for any triangle, height (h) = b sin A (if c is considered to be the base), therefore Area= ½ c b sinA

  10. 6.2 Law of Cosines • Objectives • Use the Law of Cosines to solve oblique triangles • Solve applied problems using the Law of Cosines • Use Heron’s formula to find the area of a triangle.

  11. What if know a=4”,b=6”,C=70 degrees? • Law of Sines does NOT apply • Law of Cosines was developed: • Use this to solve for c in the given triangle: • b=c, so C = B = 70 degrees, thus A = 40 degrees

  12. Using Law of Cosines, you can solve a triangle with 3 given sides • If the 3 sides are given, only 1 such triangle exists

  13. Heron’s Formula for Area of a Triangle • If the 3 sides of a triangle are known, the area can be found (based on Law of Cosines):

  14. 6.3 Polar Coordinates • Objectives • Plot points in the polar coordinate system • Find multiple sets of polar coordinate for a given point • Convert a point from polar to rectangular coordinates • Convert a point from rectangular to polar coordinates • Convert an equation from rectangular to polar coordinates • Convert an equation from polar to rectangular coordinates

  15. Defining points in the polar system • Location of a point is based on radius (distance from the origin) and theta (the angle the radius moves from standard position (positive x-axis in a cartesian system)) • Any point can be described many ways. i.e. 2 units out moving pi/2 is the same as a radius of 2 units moving -3pi/2 or a radius of 2 units moving 5pi/2

  16. What is the relationship between cartesian coordinates & polar ones? • The radius = r is the hypotenuse of a rt. triangle that has base = x & height=y • Thus, • If x = horizontal leg & y = vertical leg of a right triangle, then

  17. Find rectangular coordinates for

  18. Convert a rectangular equation to a polar equation

  19. 6.4 Graphs of Polar Equations • Objectives • Use point plotting to graph polar equations • Use symmetry to graph polar equations

  20. Graphing by Point Plotting • Given a function, in polar coordinates, you can find corresponding values for “r” and “theta” that will make your equation true. • Plotting several points and connecting the points with a curve provides the a graph of the function. • Example next page

  21. Example: • Put values in for theta that range from 0 to 2pi (once around the circle..after that the values begin repeating)

  22. Continued example

  23. “Special” curves generated by general forms

  24. r=3cos(theta)

  25. r=2+3sin(theta)

  26. r=3cos(5theta)

  27. 6.5 Complex Numbers in Polar Form: DeMoivre’s Theorem • Objectives • Plot complex numbers in the complex plane • Fine absolute value of a complex # • Write complex # in polar form • Convert a complex # from polar to rectangular form • Find products & quotients of complex numbers in polar form • Find powers of complex # in polar form • Find roots of complex # in polar form

  28. Complex number = z = a + bi • a is a real number • bi is an imaginary number • Together, the sum, a+bi is a COMPLEX # • Complex plane has a real axis (horizontal) and an imaginary axis (vertical) • 2 – 5i is found in the 4th quadrant of the complex plane (horiz = 2, vert = -5) • Absolute value of 2 – 5i refers to the distance this pt. is from the origin (continued)

  29. Find the absolute value • Since the horizontal component = 2 and vertical = -5, we can consider the distance to that point as the same as the length of the hypotenuse of a right triangle with those respective legs

  30. Expressing complex numbers in polar form • z = a + bi

  31. Express z = -5 + 3i in complex form

  32. Product & Quotient of complex numbers

  33. Multiplying complex numbers together leads to raising a complex number to a given power • If r is multiplied by itself n times, it creates • If the angle, theta, is added to itself n times, it creates the new angle, (n times theta) • THUS,

  34. Taking a root (DeMoivre’s Theorem) • Taking the nth root can be considered as raising to the (1/n)th power • Now finding the nth root of a complex # can be expressed easily in polar form • HOWEVER, there are n nth roots for any complex number & they are spaced evenly around the circle. • Once you find the 1st root, to find the others, add 2pi/n to theta until you complete the circle

  35. If you’re working with degrees add 360/n to the angle measure to complete the circle. • Example: Find the 6th roots of z= -2 + 2i • Express in polar form, find the 1st root, then add 60 degrees successively to find the other 5 roots.

  36. 6.6 Vectors • Objectives • Use magnitude & direction to show vectors are equal • Visualize scalar multiplication, vector addition, & vector subtraction as geometric vectors • Represent vectors in the rectangular coordinate system • Perform operations with vectors in terms of i & j • Find the unit vector in the direction of v. • Write a vector in terms of its magnitude & direction • Solve applied problems involving vectors

  37. Vectors have length and direction • Vectors can be represented on the rectangular coordinate system • Vectors have a horizontal & a vertical component • A vector starting at the origin and extending left 2 units and down 3 units is given below, along with the magnitude of the vector (distance from the origin):

  38. Adding & subtracting vectorsScalar Multiplication • Add (subtract) horizontal components together & add (subtract) vertical components together • v = 3i + 7j, w = -i + 2j, v – w = 4i + 5j • Scalar Multiplication: multiply the i & j components by the constant • v = 3i + 7j, 4v = 12i + 28j (the new vector is 4 times as long as the original vector)

  39. Unit Vector has the same direction as a given vector, but is 1 unit long • Unit vector = (original vector)/length of vector • Simply involves scalar multiplication once the length of the vector is determined (recall the length = length of hypotenuse if legs have lengths = a & b) • Given vector, v = -2i + 7j, find the unit vector:

  40. Writing a Vector in terms of its Magnitude & Direction • v is a nonzero vector. The vector makes an angle measured from the positive x-axis to v, and we can talk about the magnitude & direction angle of this vector:

  41. Velocity Vector: vector representing speed & direction of object in motion • Example: The wind is blowing 30 miles per hour in the direction N20 degrees E. Express its velocity as a vector v. • If the wind is N20 degrees E, it’s 70 degrees from the positive x-axis, so the angle=70 degrees and the magnitude is 30 mph.

  42. Resultant Force • Adding 2 force vector together • Add horizontal components together, add vertical components together

  43. 6.7 Dot Product • Objectives • Find dot product of 2 vectors • Find angle between 2 vectors • Use dot product to determine if 2 vectors are orthogonal • Find projection of a vector onto another vector • Express a vector as the sum of 2 orthogonal vectors • Compute work.

  44. Definition of Dot Product • The dot product of 2 vectors is the sum of the products of their horizontal components and their vertical components

  45. Find the dot product of v&w if v=3i+j and w= -2i - j • 7 • -5 • -7 • -4

  46. Properties of Dot Product • If u,v, & w are vectors and c is scalar, then

  47. Angle between vectors, v and w

  48. Parallel Vectors • Parallel: the angle between the vectors is either 0 (the vectors on top of each other) or 180 (vectors are in opposite directions), in either case, cos(0)=1, cos(180) = -1, this will be true if the dot product of v & w = (plus/minus)product of their magnitudes

  49. Orthogonal Vectors • Vectors that are perpendicular to each other. The angle between vectors is 90 degrees or 270 degrees. cos(90)=cos(270)=0 • Since they are orthogonal if the numerator = 0, thus the dot products of the 2 vectors = 0 if they are orthogonal

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