Chapter 5: Trigonometric Function
Chapter 5.2: Graph of Reciprocal Trigonometric Function
csc(x)=1/sinx
sec(x)=1/cosx
cot(x)=1/tanx
Inverse trigonometric function
〖cos〗^(-1 ) x is asking you to find an angle that has a cos ratio equal to x
Example:
〖cos〗^(-1 ) (1/2)=π/3
〖sin〗^(-1 ) x is asking you to find an angle that has a sine ratio equal to x
Example:
〖sin〗^(-1 ) (√3/2)= π/3
〖tan〗^(-1 ) x is asking you to find an angle that has tan ratio equal to x
Example:
〖tan〗^(-1 ) (1)=π/4
Chapter 5.3: Sinusiodal Functions of the form
General form of transformation of a sin or cosine function f(x) to g(x) :
f(x)= sin x → g(x)= a sin[k(x - d)] + c
f(x)= cos x → g(x)= a cos[k(x - d)] + c
Chapter 5.1: Graph of sine, cosine, and tangent functions
Graph of sine function
Graph of cosine function
Graph of tangent function
Transfromed Trigonometric Function
y=a sin〖k(x-d)+c〗
a=change in amplitude(vertical stretch/vertical compression)
k=change in period (Horizontal stretch/Horizontal compression) P=2π/k
d=Horizonal shift/phase shift
c=vertical shift
Chapter 5.5: Making Connections and Instantaneous Rate of Change
In this sub topic, we learn how to apply what we have learn previously into real-world phenomena.
Determining instantaneous rates of change for a sinusoidal function:
• By using graphing calculator, graph the function given. Use the tangent operation to find the instantaneous rate of change.
• By using formula, use the average rate of change formula to determine the instantaneous rate.
5.4 Solve Trigonometric Equations
Solving Trigonometric Equations is actually very simple. There are two important things that must always be kept in your mind when you solve Trigonometric Equations, they are:
1. The CAST rule.
1. The CAST rule.
According to the CAST rule, any angles in the first quadrant will be positive. ONLY sine will be positive on the second quadrant. ONLY tangent will be positive on the third quadrant. And ONLY cosine will be positive on fourth quadrant.
Secondly, to find the angle on the 2nd, 3rd and 4th quadrant, you must remember these few things:
2nd quadrant: π-θ
3rd quadrant: π+ θ
4th quadrant: 2π-θ
2. Special angles.
2. Special Angles
Most of the angles, you can use your calculator and use the shift sin/cos/tan function to find your answers.
However, for special angles sometimes the calculator can mislead you. So it is best to memorize the special angles.
In case you forgot the special angles, here are they:
π/3,π/6,π/4
Please remember that the CAST rule applies for special angles also.
Examples:
1. Without special angle:
Cos x = 0.45
Shift cos 0.45
X= 1.10
Cos is also positive in 4th quadrant
X= 1.10 + π
= 4.24
2. With special angle:
Sin x = ½
x= π/6
Sin is also positive in the 2nd quadrant
x=π-π/6
=5π/6
3. With 2x
Sin2x = 0.55
Shift sin 0.55
2X = 0.58, 2.56 ( π-0.58)
X= 0.29, 1.28