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8.1 The Law of Sines Congruency and Oblique Triangles ▪ Derivation of the Law of Sines ▪ Using the Law of Sines ▪ Ambiguous Case ▪ Area of a Triangle Congruence Axioms Side-Angle-Side ( ) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent. Angle-Side-Angle ( ) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent. Side-Side-Side ( ) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent. Oblique Triangles Oblique triangle: A triangle that is not a ____________ triangle The measures of the three sides and the three angles of a triangle can be found if at least one side and any other two measures are known. Data Required for Solving Oblique Triangles Case 1 Case 2 One side and two angles are known (SAA or ASA). Two sides and one angle not included between the two sides are known (SSA). This case may lead to more than one triangle. Case 3 Two sides and the angle included between the two sides are known (SAS). Case 4 Three sides are known (SSS). Note - If three angles of a triangle are known, unique side lengths cannot be found because AAA assures only similarity, not congruence. Derivation of the Law of Sines Let h be the length of the perpendicular from vertex B to side AC (or its extension) of an oblique triangle. Then c is the hypotenuse of right triangle ABD, and a is the hypotenuse of right triangle BDC. In triangle ADB, In triangle BDC, Law of Sines In any triangle ABC, with sides a, b, and c, Example 1 USING THE LAW OF SINES TO SOLVE A TRIANGLE (SAA) Solve triangle ABC if A = 32.0°, C = 81.8°, and a = 42.9 cm. Example 2 USING THE LAW OF SINES IN AN APPLICATION (ASA) Jerry wishes to measure the distance across the Big Muddy River. He determines that C = 112.90°, A = 31.10°, and b = 347.6 ft. Find the distance a across the river. Example 3 USING THE LAW OF SINES IN AN APPLICATION (ASA) Two ranger stations are on an east-west line 110 mi apart. A forest fire is located on a bearing N 42° E from the western station at A and a bearing of N 15° E from the eastern station at B. How far is the fire from the western station? Description of the Ambiguous Case If the lengths of two sides and the angle opposite one of them are given (Case 2, SSA), then zero, one, or two such triangles may exist. Applying the Law of Sines 1. 2. 3. For any angle θ of a triangle, 0 < sin θ ≤ 1. If sin θ = 1, then θ = 90° and the triangle is a right triangle. sin θ = sin(180° – θ) (Supplementary angles have the same sine value.) The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-value angle is opposite the intermediate side (assuming the triangle has sides that are all of different lengths). Example 4 SOLVING THE AMBIGUOUS CASE (NO SUCH TRIANGLE) Solve triangle ABC if B = 55°40′, b = 8.94 m, and a = 25.1 m. Note - In the ambiguous case, we are given two sides and an angle opposite one of the sides (SSA). Example 5 SOLVING THE AMBIGUOUS CASE (TWO TRIANGLES) Solve triangle ABC if A = 55.3°, a = 22.8 ft, and b = 24.9 ft. Number of Triangles Satisfying the Ambiguous Case (SSA) Let sides a and b and angle A be given in triangle ABC. (The law of sines can be used to calculate the value of sin B.) 1. If applying the law of sines results in an equation having sin B > 1, then no triangle satisfies the given conditions. 2. If sin B = 1, then one triangle satisfies the given conditions and B = 90°. 3. If 0 < sin B < 1, then either one or two triangles satisfy the given conditions. (a) If sin B = k, then let B1 = sin–1 k and use B1 for B in the first triangle. (b) Let B2 = 180° – B1. If A + B2 < 180°, then a second triangle exists. In this case, use B2 for B in the second triangle. Example 6 SOLVING THE AMBIGUOUS CASE (ONE TRIANGLE) Solve triangle ABC given A = 43.5°, a = 10.7 in., and c = 7.2 in. Example 7 ANALYZING DATA INVOLVING AN OBTUSE ANGLE Without using the law of sines, explain why A = 104°, a = 26.8 m, and b = 31.3 m cannot be valid for a triangle ABC. Area of a Triangle (SAS) In any triangle ABC, the area A is given by the following formulas: Example 8 FINDING THE AREA OF A TRIANGLE (SAS) Find the area of triangle ABC. Example 9 FINDING THE AREA OF A TRIANGLE (ASA) Find the area of triangle ABC if A = 24°40′, b = 27.3 cm, and C = 52°40′. 8.2 The Law of Cosines Derivation of the Law of Cosines ▪ Using the Law of Cosines ▪ Heron’s Formula for the Area of a Triangle Triangle Side Length Restriction In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Law of Cosines In any triangle, with sides a, b, and c, Note - If C = 90°, then cos C = 0, and the formula becomes: Example 1 USING THE LAW OF COSINES IN AN APPLICATION (SAS) A surveyor wishes to find the distance between two inaccessible points A and B on opposite sides of a lake. While standing at point C, she finds that AC = 259 m, BC = 423 m, and angle ACB measures 132°40′. Find the distance AB. Example 2 USING THE LAW OF COSINES TO SOLVE A TRIANGLE (SAS) Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m. Caution - If we used the law of sines to find C rather than B, we would not have known whether C is equal to 81.7° or its supplement, 98.3°. Example 3 USING THE LAW OF COSINES TO SOLVE A TRIANGLE (SSS) Solve triangle ABC if a = 9.47 ft, b = 15.9 ft, and c = 21.1 ft. Example 4 DESIGNING A ROOF TRUSS (SSS) Find the measure of angle B in the figure. Heron’s Area Formula (SSS) If a triangle has sides of lengths a, b, and c, with semiperimeter then the area of the triangle is Heron of Alexandria Aeolipile Example 5 USING HERON’S FORMULA TO FIND AN AREA (SSS) The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of Earth.) 8.3 Vectors, Operations, and the Dot Product Basic Terminology ▪ Algebraic Interpretation of Vectors ▪ Operations with Vectors ▪ Dot Product and the Angle Between Vectors Scalar: The _____________________ of a quantity. It can be represented by a real number. A vector in the plane is a directed line segment. Consider vector OP or __________ O is called the initial point P is called the terminal point Magnitude: length of a vector, expressed as __________ Basic Terminology Two vectors are equal if and only if they have the same __________________ and same _________________. Vectors OP and PO have the same magnitude, but opposite directions. |OP| = |PO| A=B C=D A≠E A≠F Sum of Two Vectors The sum of two vectors is also a vector. The vector sum A + B is called the resultant. The sum of a vector v and its opposite –v has magnitude 0 and is called the zero vector. To subtract vector B from vector A, find the vector sum A + (–B). Scalar Product of a Vector The scalar product of a real number k and a vector u is the vector k ∙ u, with magnitude |k| times the magnitude of u. Algebraic Interpretation of Vectors A vector with its initial point at the origin is called a ______________________ A position vector u with its endpoint at the point (a, b) is written __________ The numbers a and b are the horizontal and vertical components of vector u. The positive angle between the x-axis and a position vector is the direction angle for the vector. Magnitude and Direction Angle of a Vector a, b The magnitude (length) of a vector u = a, b is given by: The direction angle θ satisfies _______________where a ≠ 0. Example 1 FINDING MAGNITUDE AND DIRECTION ANGLE Find the magnitude and direction angle for u = 3, –2. Horizontal and Vertical Components The horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle θ are given by Example 2 FINDING HORIZONTAL AND VERTICAL COMPONENTS Vector w has magnitude 25.0 and direction angle 41.7°. Find the horizontal and vertical components. Example 3 WRITING VECTORS IN THE FORM a, b Write each vector in the figure in the form a, b. Properties of Parallelograms 1. A parallelogram is a quadrilateral whose opposite sides are parallel. 2. The opposite sides and opposite angles of a parallelogram are equal, and adjacent angles of a parallelogram are supplementary. 3. The diagonals of a parallelogram bisect each other, but do not necessarily bisect the angles of the parallelogram. Example 4 FINDING THE MAGNITUDE OF A RESULTANT Two forces of 15 and 22 newtons act on a point in the plane. (A newton is a unit of force that equals .225 lb.) If the angle between the forces is 100°, find the magnitude of the resultant vector. Vector Operations For any real numbers a, b, c, d, and k, Example 5 PERFORMING VECTOR OPERATIONS Let u = –2, 1 and v = 4, 3. Find the following. (a) u+v (b) –2u (c) 4u – 3v Unit Vectors A unit vector is a vector that has magnitude 1. i = 1, 0 j = 0, 1 Any vector a, b can be expressed in the form ai + bj using the unit vectors i and j. Dot Product The dot product (or inner product) of the two vectors u = a, b and v = c, d is denoted u ∙ v, read “u dot v,” and is given by Example 6 Find each dot product. (a) 2, 3 ∙ 4, –1 (b) 6, 4 ∙ –2, 3 FINDING DOT PRODUCTS Properties of the Dot Product For all vectors u, v, and w and real number k, (a) u ∙ v = v ∙ u (b) u ∙ (v + w) = u ∙ v + u ∙ w (c) (u + v) ∙ w = u ∙ w + v ∙ w (d) (ku) ∙ v = k(u ∙ v) = u ∙ kv (e) 0 ∙ u = 0 (f) u ∙ u = |u|2 Geometric Interpretation of the Dot Product If θ is the angle between the two nonzero vectors u and v, where 0° ≤ θ ≤ 180°, then Example 7 FINDING THE ANGLE BETWEEN TWO VECTORS Find the angle θ between the two vectors u = 3, 4 and v = 2, 1. Dot Products For angles θ between 0° and 180°, cos θ is positive, 0, or negative when θ is less than, equal to, or greater than 90°, respectively. Note - If a ∙ b = 0 for two nonzero vectors a and b, then cos θ = 0 and θ = 90°. Thus, a and b are perpendicular or ________________________ vectors. 8.4 Applications of Vectors The Equilibrant ▪ Incline Applications ▪ Navigation Applications Sometimes it is necessary to find a vector that will counterbalance a resultant. This opposite vector is called the ____________________. The equilibrant of vector u is the vector –u. Example 1 FINDING THE MAGNITUDE AND DIRECTION OF AN EQUILIBRANT Find the magnitude of the equilibrant of forces of 48 newtons and 60 newtons acting on a point A, if the angle between the forces is 50°. Then find the angle between the equilibrant and the 48-newton force. Example 2 FINDING A REQUIRED FORCE Find the force required to keep a 50-lb wagon from sliding down a ramp inclined at 20° to the horizontal. (Assume there is no friction.) Example 3 FINDING AN INCLINE ANGLE A force of 16.0 lb is required to hold a 40.0 lb lawn mower on an incline. What angle does the incline make with the horizontal? Example 4 APPLYING VECTORS TO A NAVIGATION PROBLEM A ship leaves port on a bearing of 28.0° and travels 8.20 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port? Airspeed and Groundspeed The airspeed of a plane is its speed relative to the air. The groundspeed of a plane is its speed relative to the ground. The groundspeed of a plane is represented by the vector sum of the airspeed and windspeed vectors. Example 5 APPLYING VECTORS TO A NAVIGATION PROBLEM A plane with an airspeed of 192 mph is headed on a bearing of 121°. A north wind is blowing (from north to south) at 15.9 mph. Find the groundspeed and the actual bearing of the plane. 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients The Complex Plane and Vector Representation ▪ Trigonometric (Polar) Form ▪ Fractals ▪ Products of Complex Numbers in Trigonometric Form ▪ Quotients of Complex Numbers in Trigonometric Form Horizontal axis: real axis Vertical axis: imaginary axis Each complex number a + bi determines a unique position vector with initial point (0, 0) and terminal point (a, b). The sum of two complex numbers is represented by the vector that is the resultant of the vectors corresponding to the two numbers. (4 + i) + (1 + 3i) = 5 + 4i Example 1 EXPRESSING THE SUM OF COMPLEX NUMBERS GRAPHICALLY Find the sum of 6 – 2i and –4 – 3i. Graph both complex numbers and their resultant. Relationships Among x, y, r, and θ. Trigonometric (Polar) Form of a Complex Number The expression r(cos θ + i sin θ) is called the trigonometric form (or polar form) of the complex number x + yi. The expression cos θ + i sin θ is sometimes abbreviated cis θ. Using this notation, r(cos θ + i sin θ) is written r cis θ. The number r is the absolute value (or modulus) of x + yi, and θ is the argument of x + yi. Example 2 CONVERTING FROM TRIGONOMETRIC FORM TO RECTANGULAR FORM Express 2(cos 300° + i sin 300°) in rectangular form. Converting From Rectangular Form to Trigonometric Form Step 1 Step 2 Sketch a graph of the number x + yi in the complex plane. Find r by using the equation Step 3 Find θ by using the equation choosing the quadrant indicated in Step 1. Caution Be sure to choose the correct quadrant for θ by referring to the graph sketched in Step 1. Example 3(a) CONVERTING FROM RECTANGULAR FORM TO TRIGONOMETRIC FORM Write in trigonometric form. (Use radian measure.) Example 3(b) CONVERTING FROM RECTANGULAR FORM TO TRIGONOMETRIC FORM Write –3i in trigonometric form. (Use degree measure.) CONVERTING BETWEEN TRIGONOMETRIC AND Example 4 RECTANGULAR FORMS USING CALCULATOR APPROXIMATIONS Write each complex number in its alternative form, using calculator approximations as necessary. (a) 6(cos 115° + i sin 115°) (b) 5 – 4i Example 5 DECIDING WHETHER A COMPLEX NUMBER IS IN THE JULIA SET The figure shows the fractal called the Julia set. To determine if a complex number z = a + bi belongs to the Julia set, repeatedly compute the values of: If the absolute values of any of the resulting complex numbers exceed 2, then the complex number z is not in the Julia set. Otherwise z is part of this set and the point (a, b) should be shaded in the graph. Determine whether each number belongs to the Julia set. Product Theorem are any two complex numbers, then: In compact form, this is written: Example 6 USING THE PRODUCT THEOREM Find the product of 3(cos 45° + i sin 45°) and 2(cos 135° + i sin 135°). Write the result in rectangular form. Quotient Theorem are any two complex numbers, where In compact form, this is written Example 7 Find the quotient USING THE QUOTIENT THEOREM Write the result in rectangular form. 8.6 De Moivre’s Theorem; Powers and Roots of Complex Numbers Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of Complex Numbers De Moivre’s Theorem is a complex number, then In compact form, this is written Example 1 Find FINDING A POWER OF A COMPLEX NUMBER and express the result in rectangular form. nth Root For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if nth Root Theorem If n is any positive integer, r is a positive real number, and θ is in degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly n distinct nth roots, given by where Note - In the statement of the nth root theorem, if θ is in radians, then Example 2 FINDING COMPLEX ROOTS Find the two square roots of 4i. Write the roots in rectangular form. Example 3 Find all fourth roots of FINDING COMPLEX ROOTS Write the roots in rectangular form. Example 3 FINDING COMPLEX ROOTS (continued) The graphs of the roots lie on a circle with center at the origin and radius 2. The roots are equally spaced about the circle, 90° apart. Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS Find all complex number solutions of x5 – i = 0. Graph them as vectors in the complex plane. Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS (continued) The graphs of the roots lie on a unit circle. The roots are equally spaced about the circle, 72° apart. 8.7 Polar Equations and Graphs Polar Coordinate System ▪ Graphs of Polar Equations ▪ Converting from Polar to Rectangular Equations ▪ Classifying Polar Equations The polar coordinate system is based on a point, called the pole, and a ray, called the polar axis. Point P has rectangular coordinates (x, y). Point P can also be located by giving the directed angle θ from the positive x-axis to ray OP and the directed distance r from the pole to point P. The polar coordinates of point P are (r, θ). If r > 0, then point P lies on the terminal side of θ. If r < 0, then point P lies on the ray pointing in the opposite direction of the terminal side of θ, a distance |r| from the pole. Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r, ), then these coordinates are related as follows. Example 1 PLOTTING POINTS WITH POLAR COORDINATES Plot each point by hand in the polar coordinate system. Then, determine the rectangular coordinates of each point. (a) P(2, 30°) Example 1 PLOTTING POINTS WITH POLAR COORDINATES Example 1 PLOTTING POINTS WITH POLAR COORDINATES (continued) While a given point in the plane can have only one pair of rectangular coordinates, this same point can have an infinite number of pairs of polar coordinates. Example 2 GIVING ALTERNATIVE FORMS FOR COORDINATES OF A POINT (a) Give three other pairs of polar coordinates for the point P(3, 140°). (b) Determine two pairs of polar coordinates for the point with the rectangular coordinates (–1, 1). Example 3 EXAMINING POLAR AND RECTANGULAR EQUATION OF LINES AND CIRCLES For each rectangular equation, give the equivalent polar equation and sketch its graph. (a) y = x – 3 Example 3 (b) EXAMINING POLAR AND RECTANGULAR EQUATION OF LINES AND CIRCLES (continued) Example 4 GRAPHING A POLAR EQUATION (CARDIOID) Example 5 GRAPHING A POLAR EQUATION (ROSE) Example 6 Graph . GRAPHING A POLAR EQUATION (LEMNISCATE) Example 7 GRAPHING A POLAR EQUATION (SPIRAL OF ARCHIMEDES) Graph r = 2θ, (θ measured in radians). Example 8 Convert the equation CONVERTING A POLAR EQUATION TO A RECTANGULAR EQUATION to rectangular coordinates and graph. Classifying Polar Equations Circles and Lemniscates Limaçons Classifying Polar Equations Rose Curves 2n leaves if n is even, n≥2 n leaves if n is odd