f(x) = a sin [k(x-d)] + c
f(x)= a cos [k(x-d)] + c
|a| = amplitude
k = horizontal / vertical compression
d = phase shift
c = vertical translation
*** Period = 2π/k
Chapter 5: Trigonometric Function
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:
2. Special angles.
Examples:
3. With 2x
Sin2x = 0.55
Shift sin 0.55
2X = 0.58, 2.56 ( π-0.58)
X= 0.29, 1.28
2. With special angle:
Sin x = ½
x= π/6
Sin is also positive in the 2nd quadrant
x=π-π/6
=5π/6
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. 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.
1. The CAST rule.
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
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.
Chapter 5.5: Making Connections and Instantaneous Rate of Change
Determining instantaneous rates of change for a sinusoidal function:
• By using formula, use the average rate of change formula to determine the instantaneous rate.
• By using graphing calculator, graph the function given. Use the tangent operation to find the instantaneous rate of change.
In this sub topic, we learn how to apply what we have learn previously into real-world phenomena.
Chapter 5.1: Graph of sine, cosine, and tangent functions
Transfromed Trigonometric Function
y=a sin〖k(x-d)+c〗
c=vertical shift
d=Horizonal shift/phase shift
k=change in period (Horizontal stretch/Horizontal compression) P=2π/k
a=change in amplitude(vertical stretch/vertical compression)
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)= cos x → g(x)= a cos[k(x - d)] + c
f(x)= sin x → g(x)= a sin[k(x - d)] + c
Chapter 5.2: Graph of Reciprocal Trigonometric Function
Inverse trigonometric function
〖tan〗^(-1 ) x is asking you to find an angle that has tan ratio equal to x
Example:
〖tan〗^(-1 ) (1)=π/4
〖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
〖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
cot(x)=1/tanx
sec(x)=1/cosx
csc(x)=1/sinx
