Ch. 6 Additional Topics in Trig

6.1 Law of Sines

A. OBJ: to use the Law of Sines to solve for an oblique triangle, to use the Law to find area of a triangle.

B. FACTS:

1. Law of Sines:

sin Aa

= sin Bb

= sin Cc

2. Area of an Oblique Triangle:

A = ½ bc Sin A = ½ ac Sin B = ½ ab Sin C

6.2 Law of Cosines

A. OBJ: to find the missing sides and angles of an oblique triangle using law of cosines.

B. FACTS:

1. a2 = b2 + c2 – 2bcCosA

b2 = a2 + c2 – 2acCosB

c2 = a2 + b2 – 2abCosC

2. Area = √s (s−a)(s−b)(s−c )

S = 1//2 (a+b+c)

6.3 Vectors in the Plane

A. OBJ: to represent vectors as line segments with direction and magnitude. To perform operations with vectors,

B. FACTS:1. Vector – line segment with direction and magnitude2. Magnitude – length = distance formula - || PQ ||3. Direction – initial pt. P and terminal pt. Q4. Component form :

5. Scalar multiplication:

6. Addition/Subtraction:

7. Properties of Vectors:

8. Direction of Vectors;

9. Using Trig to find θ:

If v = ai + bj is any vector that makes an angle with the positive x-axis, then it has the same direction as u and youcan write

v = || v ||cos , sin

= || v ||(cos )i + || v ||(sin )j.

AND it follows that the direction angle for v is determined from

If u is a unit vector such that is the angle (measured counterclockwise) from the positive x-axis to u, then the terminal point of u lies on the unit circle and you haveu = x, y = cos , sin = (cos )i + (sin )j

The angle is the direction angle of the vector u.

6.4 Vectors and Dot Products

A. OBJ: to find the dot product of 2 vectors. To find the < between 2 vectors. To write a vector as the sum of 2 vector components.

B. FACTS:1. Dot product is a 3rd vector operation that gives you a scalar.

2.

3. The angle between two nonzero vectors is the angle , 0 ,between their respective standard position vectors is found using dot product. u v = || u || || v || cos

4. The five possible orientations of two vectors are shown below.

5. Orthogonal Vector = perpendicular = intersect at rt. <

6.5 Trig form of a Complex Number

A. OBJ: to write trig form of complex numbers. To plot complex numbers and perform operations using trig.

B. FACTS:

1. complex plane:

2. complex number: a ± bi

3. absolute value is the distance between (0, 0) and (a, b):

4. Trig form of a complex number = Polar form:

x = a = r cos and y = b = r sin , where

Consequently, you have a + bi = (r cos ) + (r sin )i

5. Product/Quotient:

6. Power of Complex Number:

7. Root of a Complex Number: