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Applications of Trigonometry Chapter 7

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.1 The Law of Sines  Use the law of sines to solve triangles.  Find the area of any triangle given the lengths of two sides and the measure of the included angle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Oblique Triangles To solve a triangle means to find the lengths of all its sides and the measures of all its angles. Any triangle that is not a right triangle is called oblique. Any triangle, right or oblique, can be solved if at least one side and any other two measures are known.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Possible Triangles

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Law of Sines In any triangle ABC, Thus in any triangle, the sides are proportional to the sines of the opposite angles. B A C a c b

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a Triangle (AAS) In triangle ABC, A = 57 , B = 43 , and b = Solve the triangle. C = (180°  (57° + 43°)) C = 180°  100° = 80° Find a and c by using the law of sines: B A C a c  57 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving a Triangle (SSA) If the given angle is acute, then there may be: a) no solution, b) one solution, or c) two solutions If the given angle is obtuse, then there may be: a) no solution, or b) one solution

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example In triangle ABC, b = 8.6, c = 6.2, and C = 35 . Solve the triangle. Solution: C B A?A? a 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued There are two solutions: or

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Area of a Triangle The area K of any  ABC is one half the product of the lengths of two sides and the sine of the included angle:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the area of the triangle, ABC with A = 72 , b = 16, and c = 10. Solution: 10 A B C 

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.2 The Law of Cosines  Use the law of cosines to solve triangles.  Determine whether the law of sines or the law of cosines should be applied to solve a triangle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In any triangle ABC, Thus, in any triangle, the square of a side is the sum of the squares of the other two sides, minus twice the product of those sides and the cosine of the included angle. When the included angle is 90 , the law of cosines reduces to the Pythagorean theorem. The Law of Cosines C B A b a c

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Triangles (SAS) Solve  ABC if a = 4, c = 6, and B = . A C B b  4

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Triangles (SSS) Solve  ABC if a = 15, b = 11, and c = 8. Solution: Solve for A first A B C

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Triangles (SSS) continued A B C

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.3 Complex Numbers: Trigonometric Form  Graph complex numbers.  Given a complex number in standard form, find trigonometric, or polar, notation; and given a complex number in trigonometric form, find standard notation.  Use trigonometric notation to multiply and divide complex numbers.  Use DeMoivre’s theorem to raise complex numbers to powers.  Find the nth roots of a complex number.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Value of a Complex Number The absolute value of a complex number a + bi is Example: Find the absolute value of: a) 4 + 5i b)  1  i

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Trigonometric Notation for Complex Numbers a + bi = r(cos  + i sin  ) Example: Find trigonometric notation for  1  i. Solution: First, find r. Thus,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Complex Numbers: Multiplication For any complex numbers

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply and express in standard notation.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Complex Numbers: Division For any complex numbers

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Divide and express in standard notation.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DeMoivre’s Theorem For any complex number any natural number n,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find (  1  i) 5. Solution: First, find trigonometric notation for  1  i. Then

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Roots of Complex Numbers The nth roots of a complex number, are given by where k = 0, 1, 2, …, n  1.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the square roots of. Trigonometric notation: For k = 0, the root is For k = 1, the root is

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.4 Polar Coordinates and Graphs  Graph points given their polar coordinates.  Convert from rectangular to polar coordinates and from polar to rectangular coordinates.  Convert from rectangular to polar equations and from polar to rectangular equations.  Graph polar equations.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Coordinates Any point has rectangular coordinates (x, y) and polar coordinates (r,  ). To plot points on a polar graph: Locate the directed angle . Move a directed distance r from the pole. If r > 0, move along ray OP. If r < 0, move in the opposite direction of ray OP.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph each of the following points. a)A(3, 60  ) b)B(0, 10  ) c)C(  5, 120  ) d)D(1,  60  ) e)E f)F

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Conversion Formulas

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Convert (4, 2) to polar coordinates. Thus (r,  ) =

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Convert to rectangular coordinates. (x, y) =

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar and Rectangular Equations Some curves have simpler equations in polar coordinates than in rectangular coordinates. For others, the reverse is true. Convert x + 2y = 10 into a polar equation. x + 2y = 10

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Convert r =  3 cos   sin  into a rectangular equation.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphing Polar Equations Graph r =  2 sin  r  r  r 

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphing with graphing calculator Graph the equation r + 2 = 2 sin 2 . Solution solve for r.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.5 Vectors and Applications  Determine whether two vectors are equivalent.  Find the sum, or resultant, of two vectors.  Resolve a vector into its horizontal and vertical components.  Solve applied problems involving vectors.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Vector A vector in the plane is a directed line segment. Two vectors are equivalent if they have the same magnitude and direction. Consider a vector drawn from point A to point B A is called the initial point B is called the terminal point Equivalent vectors: vectors with the same length and direction. Magnitude: length of a vector, expressed as

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the magnitude of the vector, v, with initial point (  1, 2) and terminal point (3,  2).

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Vector Addition The sum of two vectors is called the resultant. If

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallelogram Law Vector addition is commutative.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant that it makes with the larger force. The resultant vector, v, has magnitude 51 and makes an angle of 11.3  with the larger force  v

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Components Resolving a vector into its vector components—write a vector, w, as a sum of two vectors, u and v, which are the components. Most often, the two components will be the horizontal component and the vertical component of the vector.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A vector w has a magnitude of 45 and rests on an incline of 20 . Resolve the vector into its horizontal and vertical components. The horizontal component of w is 42.3 right and the vertical component of w is 15.4 up. v u 

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 7.6 Vector Operations  Perform calculations with vectors in component form.  Express a vector as a linear combination of unit vectors.  Express a vector in terms of its magnitude and its direction.  Solve applied problems involving forces in equilibrium.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Position Vectors Standard position—the initial point is the origin and the terminal point is (a, b). Notation: v = a is the scalar horizontal component b is the scalar vertical component Scalar—numerical quantity rather than a vector quantity

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Component Form of a Vector The component form of with A = (x 1, y 1 ) and C = (x 2, y 2 ) is Example: Find the component form of if A = (3,  2) and B = (  5,  6). Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Length of a Vector The length, or magnitude, of a vector v = is given by Equivalent Vectors Let u = if and only if u 1 = v 1 and u 2 = v 2.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Operations on Vectors

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Operations on Vectors continued

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example a) 4v b) 2u + v c) 2u  3v d) |u + 4v|  Let u = and v =. Find each of the following.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Properties of Vector Addition and Scalar Multiplication

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Unit Vectors A vector of magnitude 1. If v is a vector and v  O, then is a unit vector in the direction of v.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find a unit vector that has the same direction as

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Combinations i = is a vector parallel to the x-axis. j = is a vector parallel to the y-axis. The linear combination, v, of unit vectors i and j, where Example: The linear combination of i and j for is  2i + 3j.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Direction Angles A vector u in standard position on the unit circle can be written as where  is called the direction angle. It is measured counterclockwise from the positive x-axis.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Determine the direction angle  of the vector w =  3i + 4j. Solution:  is a 2 nd quadrant angle, so  =  53.1   = .

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions The dot product of two vectors u = and v = (Note that u 1 v 1 + u 2 v 2 is a scalar, not a vector.) Angle Between Two Vectors If  is the angle between two nonzero vectors u and v, then

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution: Find u v, |u|, |v|. u v =  4(3) + 2(  1) =  14 Find the angle between u = and v =.