Chapter 6 Trigonometric Functions
Sinusoidal Graphs
*y= A sin(wx) + B, w>0 *y= A cos(wx) + B, w>0 *y= A sin(wx-theta) + B = A sin[w(x-theta/w)] + B *y= A cos(wx-theta) + B = A cos[w(x-theta/w)] + B
One degree
1/360 of a revolution.
Amplitude
Amplitude = |A|
Straight Angle
An angle that measures 180 degrees, or 1/2 of a revolution.
Right Angle
An angle that measures 90 degrees, or 1/4 of a revolution.
Standard Position
An angle θ is said to be in ______________ if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis.
Circular Functions
Another word for trigonometric functions.
Cosine
Associates with t the x-coordinate of P and is denoted by: cos t = x
sine
Associates with t the y-coordinate of P and is denoted by: sin t = y
One Second (1")
Defined as 1/60 minute, or 1/3600 degree.
Arc Length
For a circle of radius r, a central angle of θ radians subtends an arc whose length s is: s = rθ
1 Radian
If the radius of the circle is r and the length of the arc subtended by the central angle is also r, then the measure of the angle is __________.
Quadrantal Angle
If the terminal side of an angle θ lies on the x-axis or y-axis, θ is a ____________.
Six trigonometric functions of the angle θ
If θ = t radians, the __________ are defined as: sin θ = sin t cos θ = cos t tan θ = tan t csc θ = csc t sec θ = sec t cot θ = cot t
Initial side
One ray of an angle.
Period
Period = T = 2pi/w
Phase Shift
Phase Shift = theta/w
Linear Speed
Suppose than an object moves around a circle of radius r at a constant speed. If s is the distance traveled in time t around this circle, then the ___________ v of the object is defined as: v = s/t
Vertex
The Starting point V of a ray.
Angular Speed
The ________ ω (greek letter omega) of this object is the angle θ (measured in radians) swept out, divided by the elapsed time t; that is: ω = θ/t
Area of a Sector
The area A of the sector of a circle of radius r formed by a central angle of θ is: A = (1/2) * r^2 * θ.
Reciprocal Functions
The cosecant and secant functions; graphed using csc x=1/sinx and sec x=1/cosx.
Terminal side
The other ray of an angle.
Ray/Half-line
The portion of a line that starts at a point V on the line and extends indefinitely in one direction.
"Lies in that quadrant"
When an angle θ is in the terminal side will lie either in a quadrant, in which case we say that θ _____________, or the terminal side will lie on the x-axis or y-axis, in which case we say that θ is a quadrantal angle.
Negative Angle
When the rotation is in the clockwise direction, the angle is _________.
Positive Angle
When the rotation is in the counterclockwise direction, the angle is _________.
Angle
When two rays are drawn with a common vertex.
periodic
a function f is called periodic if there is a positive number p such that, whenever (theta) is in the domain of f, so is (theta)+ p and f((Theta)+p)= f((theta))
Central Angle
a positive angle whose vertex is at the center of a circle.
reciprocal identities
csc(theta) =1/sin(theta) sec(theta)=1/cos(theta) cot(theta)=1/tan(theta)
One Minute (1')
defined as 1/60 degree.
Theorem for Amplitude and Period of sin and cos
if w>0, the amplitude and period of y=Asin(wx) and y=Acos(wx) are given by Amplitude=IAI and Period=T=2(pi)/w
Even-Odd Properties
sin(-theta)=-sin(theta) cos(-theta)=cos(theta) tan(-theta)=-tan(theta) csc(-theta)=-csc(theta) sec(-theta)=sec(theta) cot(-theta)=-cot(theta)
fundamental identities
sin^2(theta)+cos^2(theta)=1 tan^2(theta)+1=sec^2(theta) cot^2(theta)+1=csc^2(theta)
quotient identities
tan(theta)=sin(theta)/cos(theta) cot(theta)= cos(theta)/ sin(theta)
fundamental period of f
the smallest p for f(theta+p)=f(theta)
Tangent
x cannot = 0; Associates with t the ratio of the y-coordinate to the x-coordinate of P and is denoted by: tan t = y/x
Secant
x cannot = 0; defined as: sec t = 1/x
Cotangent
y cannot = 0; defined as: cot t = x/y
Cosecant
y cannot = 0; defined as: csc t = 1/y
