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Section 7.1 & 7.2- Oblique Triangles (non-right triangle)
LAW OF SINES If ABC is a triangle with sides a, b, and c, then A, B, and C are acute A is obtuse Two angles and any side: AAS or ASA Two sides and an excluded angle: SSA (ASS)
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Find the remaining angle and sides for a triangle with the given information.
1. Find all angles first. 2. Find sides by using the proportion formula for Law of Sines, use given values! sin(49o) sin(29o) sin(102o) 28
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Ambiguous Case SSA (ASS) A is an acute angle.
Condition a < h a = h a > b # of Triangles None One One
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Ambiguous Case SSA (ASS) A is an acute angle.
Condition h < a < b # of Triangles TWO B + B’ = 180o a C’ b b a B’ A A c’
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Ambiguous Case SSA (ASS) A is an obtuse angle.
Condition a < b a > b # of Triangles NONE One
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Find the remaining angle and sides for a triangle with the given information.
Draw the triangle with the given angle in the lower left corner and solve for h. a b A NO TRIANGLE
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Find the remaining angle and sides for a triangle with the given information.
Draw the triangle with the given angle in the lower left corner and solve for h. c b C FALSE There is a triangle. FALSE More than 1 triangle. A’ b Condition h < c < b < 12 < 31 # of Triangles TWO c’ c B’ C a’
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c A’ b c C B’ c’ c b A’ b’ C c C B’ a’
Find the remaining angle and sides for a triangle with the given information. Since the inverse of sine will return an acute angle, we will solve the Acute Triangle first! Find angle B first and then angle A. c A’ b c C B’ c’ c b B’ = 180o – B A’ b’ C c sin(97.9o) sin(20o) C B’ a’ 12 31 sin(42.1o) sin(20o) 12
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Area of an Oblique triangle SAS
c Area of any triangle is one-half the product of the lengths of two sides times the sine of the included angle. Find the area for a triangle with the given information.
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Section 7.3- Oblique Triangles – Law of Cosines SAS & SSS
Find the distance of “a.” b A c Find a trig. expression for x and y. Substitute the trig. expressions for x and y.
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Alternative Form Standard Form
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Find the remaining angles and side of the triangle. SAS
1. Find the side opposite the given angle. C a b = 9 25o A B c = 12 2. Find the angle opposite the shortest given side by the Law of Sines and then subtract the two acute angles from 180o. sin(25o) 9 5.41 12
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Find the angles of the triangle. SSS
1. Use the Law of Cosines to find the angle opposite the longest side. B a = 14 c = 8 A C b = 19 Negative value, means Quad. 2 for cos-1x, obtuse angle. 2. Find either acute angle by the Law of Sines and then subtract the two angles from 180o. sin(116.8o) 14 19 8
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Heron’s Area Formula SAS & SSS
Given any triangle with sides a, b, and c, the area of the triangle is… where s = ( a + b + c )/2. C b = 53 a = 43 Find the area of the triangle. A B 1. Find the value of s. c = 72
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v Section 7.4 - Vectors in the Plane
Force and velocity involve both magnitude (distance) and direction (slope) and cannot be completely characterized by a single real number. We will use a DIRECTIONAL LINE SEGMENT (RAY) to represent force and velocity (vectors). Q Terminal Point v P Initial Point Let u represent the directed line segment from P(0,0) to Q(3,2) and v be the directed line segment from R(1,2) to S(4,4). Show they are equivalent. Equivalent vectors must have the same magnitude and direction. v u Same Magnitude Same Direction
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The multiplication of a real number k and a vector v is called scalar multiplication. We write this product kv. Multiplying a vector by any positive real number (except 1) changes the magnitude of the vector but not its direction. Multiplying a vector by any negative number reverses the direction of the vector.
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The sum of u and v, denoted u + v is called the resultant vector
The sum of u and v, denoted u + v is called the resultant vector. A geometric method for adding two vectors is shown in the figure. Here is how we find this vector: u u + v u u – v v - v v Parallelogram Law Connecting the terminal point of the first vector with the initial point of the second vector to obtain the sum of two vectors. Find u - v
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The i and j Unit Vectors The Vector i is the unit vector whose direction is along the positive x-axis. Vector j is the unit vector whose direction is along the positive y-axis. y 1 j x 1 i Consider the point P(a, b). The initial point is at (0, 0) and terminal point is at (a, b). Vector v can be represented as v = ai + bj. y Another notation is position vector form. v v = ai + bj bj x ai
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u = (-2 – 3)i + (5 – (-1))j u = – 5i + 6j u
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Adding and Subtracting Vectors in Terms of i and j
Position vector form. v = w = v + w = v – w = If v = 7i + 3j and w = 4i – 5j, find the following vectors: a. v + w b. v – w v = w = = (7i + 3j) + (4i – 5j) v + w = Comb. Like Terms = 11i – 2j Dist. Prop of minus sign = (7i + 3j) – (4i – 5j) v – w = = 7i + 3j – 4i + 5j = 3i + 8j
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Scalar Multiplication with a Vector in Terms of i and j
If v = 7i + 3j and w = 4i – 5j, find the following vectors: a. 3v b. 3v – 2w = 3(7i + 3j) v = w = = 21i + 9j 3v = = 3(7i + 3j) – 2(4i – 5j) 3v – 2w = = 21i + 9j – 8i + 10j = 13i + 19j Or distribute the minus sign
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Find the unit vector in the same direction as v = 4i – 3j
Find the unit vector in the same direction as v = 4i – 3j. Then verify that the vector has magnitude 1.
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Writing a Vector in Terms of Its Magnitude and Direction
v = ai + bj v = ai + bj v = i j Remember your identities…
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The jet stream is blowing at 60 miles per hour in the direction N45°E
The jet stream is blowing at 60 miles per hour in the direction N45°E. Express its velocity as a vector v in terms of i and j. The jet stream can be expressed in terms of i and j as v = i j
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The dot product of two vectors results in a scalar (real number) value, rather than a vector.
If v = 7i – 4j and w = 2i – j, find each of the following dot products: a. b. c. v w w v w w
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Alternative Formula for the Dot Product
Find the angle between the two vectors v = 4i – 3j and w = i + 2j. Round to the nearest tenth of a degree. The angle between the vectors is
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Two forces, F1 and F2, of magnitude 30 and 60 pounds, respectively, act on an object. The direction of F1 is N10°E and the direction of F2 is N60°E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force.
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Identities
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Parallel and Orthogonal Vectors
Two vectors are parallel when the angle between the vectors is 0° or 180°. If = 0°, the vectors point in the same direction. If = 180°, the vectors point in opposite directions. Two vectors are orthogonal when the angle between the vectors is 90°. Are the vectors v = 2i + 3j and w = 6i – 4j orthogonal? Yes v w Dot Product of Acute Angles are positive, Right Angles are zero, and Obtuse Angles are negative.
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Using Vectors to Determine Weight.
A force of 600 pounds is required to pull a boat and trailer up a ramp inclined 15o from the horizontal. Find the combined weight of the boat and trailer. B W Force of gravity = combined weight. Force against ramp. Force required to move boat up ramp = 600lbs C A
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Using Vectors. u v u + v ||u+v||
A plane is flying at a bearing N 30o W at 500 mph. At a certain point the plane encounters a 70 mph wind with the bearing N 45o E. What are the resultant speed and direction of the plane? u v 45o u + v v u 120o u + v ||u+v||
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Using Vectors to find tension.
Determine the weight of the box. A B 50o 30o 879.4 lbs 652.7 lbs u C v ??? lbs 1000 lbs u + v u + v
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