Download Ch. 6 Notes - Glassboro Public Schools

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 𝐴
𝑎
=
𝑆𝑖𝑛 𝐵
𝑏
=
𝑆𝑖𝑛 𝐶
𝑐
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 magnitude
2. Magnitude – length = distance formula - || PQ ||
3. Direction – initial pt. P and terminal pt. Q
4. Component form :
5. Scalar multiplication:
6. Addition/Subtraction:
7. Properties of Vectors:
8. Direction of Vectors;
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 have
u = x, y = cos , sin  
= (cos  )i + (sin  )j
The angle  is the direction
angle of the vector u.
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 you
can write
v = || v ||cos , sin  
= || v ||(cos  )i + || v ||(sin  )j.
AND it follows that the direction angle  for v is determined from
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: