Mathematics Dictionary
Dr. K. G. Shih
Cosine function
Symbol Defintion
Example : Sqr(x) = square root of x
Q01 |
- Cosine function : y = cos(x)
Q02 |
- Cosine Law
Q03 |
- Half angle : cos(A/2) = Sqr(s)*(s - a)/(b*c)
Q04 |
- Cosine function and sine function
Q05 |
- Locus of R = cos(A)
Q06 |
- Solve triangle if SAS is given
Q07 |
- cos(A) and sec(A)
Q08 |
- Formula : cos(A + B) and cos(A - B)
Q09 |
- Formula : series
Q01. Cosine function
Triangle definition
cos(A) = Adj/Hyp
Rectangular coordinate definition
cos(A) = x/r
r = Sqr(x^2 + y^2)
Cos(A) = (+) in 1st quadrant
Cos(A) = (-) in 2nd quadrant
Cos(A) = (-) in 3rd quadrant
Cos(A) = (+) in 4th quadrant
Properties of y = cos(x)
It is a periodic function and period = 2*pi
Five important points between 0 and 2*pi
At x = 000 : y = +1
At x = 090 : y = +0
At x = 180 : y = -1
At x = 270 : y = +0
At x = 360 : y = +1
Sketch : Five point method
Sketch y = a + b*cos(x)
Five point method
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Q02. Coine Law
Cosine law.
Defintion : Trianle ABC
a^2 = b^2 + c^2 - 2*b*c*cos(A).
b^2 = c^2 + a^2 - 2*c*a*cos(B).
c^2 = a^2 + b^2 - 2*b*c*cos(C).
Application : Solve triangle if SAS are given
Solve triangle if b, A, c are given.
Solve triangle if c, B, a are given.
Solve triangle if a, C, b are given.
Proof of cosine law
Proof of cosine law
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Q03. Half angle : cos(A/2) = Sqr(s)*(s - a)/(b*c)
Proof
Prove
cos(A/2) = Sqr(s*(s - a)/(b*c))
Half angle in terms of cos(A)
cos(A) = Sqr((1 + cos(A))/2)
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Q04. Cosine function and sine function
Relation
cos(A) = +sin(090 - A)
cos(A) = -sin(090 + A)
cos(A) = -sin(270 - A)
cos(A) = +sin(270 + A)
Pythagorean relation
cos(A)^2 + sin(A)^2 = 1
This is a unit circle in parametric equation
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Q05. Locus of R = cos(A)
Solution
Locus
of R = cos(A)
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Q06. Solve triangle if SAS are given
Example : if a = 2 and b = 3 and angle C = 60 degrees, find other side c.
c^2 = 2^2 + 3^2 - 2*2*3*cos(60).
c^2 = 4 + 9 - 12*(1/2).
c^2 = 7.0
Hence c = 2.64575.
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Q07. Cos(A) and sec(A)
Relation
sec(A) = 1/cos(A)
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Q08. Cos(A + B) and cos(A - B)
Formula
cos(A + B) = cos(A)*cos(B) - sin(A)*sin(B)
cos(A - B) = cos(A)*cos(B) + sin(A)*sin(B)
Proof
Trigonomery
1. cos(A + B) = cos(A)*cos(B) - sin(A)*sin(B)
2. cos(2*A) = 2*cos(A)^2 - 1
Proof with diagrams
Trigonomery
cos(A - B) = cos(A)*cos(B) + sin(A)*sin(B)
Proof : Usinf cosine law
Trigonomery
cos(A - B) = cos(A)*cos(B) + sin(A)*sin(B)
Product of cos(A)*cos(B)
cos(A)*cos(B) = (cos(A + B) + cos(A - B))/2
Sum of cos(A) + cos(B)
cos(A) + cos(B) = 2*(cos((A + B)/2) + cos((A - B)/2)
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Q09. Series of cos(x)
Series
cos(x) = 1 - (x^2)/2! + (x^4)/4! - .....
Derivative of y = cos(x)
y' = -sin(x)
y" = -cos(x)
e^(i*x)
e^(i*x) = cos(x) + i*sin(x)
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Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
