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)

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

Ambiguous Case SSA (ASS) A is an acute angle. Condition a < h a = h a > b # of Triangles None One One

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’

Ambiguous Case SSA (ASS) A is an obtuse angle. Condition a < b a > b # of Triangles NONE One

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

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 10.6< 12 < 31 # of Triangles TWO c’ c B’ C a’

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

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.

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.

Alternative Form Standard Form

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

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

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

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

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.

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

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

u = (-2 – 3)i + (5 – (-1))j u = – 5i + 6j u

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

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

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.

Writing a Vector in Terms of Its Magnitude and Direction v = ai + bj v = ai + bj v = i + j Remember your identities…

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

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

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

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.

Identities

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.

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

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||

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