Trigonometric Ratios in Radians and Degrees
What are the three Reciprocal Trigonometric Ratios and what are they the reciprocal of?
csc=hyp/opp >sin
sec=hyp/adj >cos
cot=adj/opp >tan
Values of Trigonometric ratios in: Quadrant 1
All values are positive and the value of theta is itself.
Identify the Pythagorean Identity and the Quotient Identity.
Pythagorean Identity- sin2|theta|+cos2|theta|=1 ;.
tan2|theta|+1=sec2|theta|
cot2|theta|+1=csc2|theta|
Quotient Identity- sin|theta|/cos|theta|=tan|theta|
What is the general equation of a trigonometric function?
f(x)=a(sin)[k(x-d)]+c
In the function f(x)=(a)sin(x), what affect does a have on the function?
*vertical reflection(reflection in the x-axis) if a<0
*vertical stretch if a>1 or a<-1
*vertical compression if -1<a<1
-this affects the the amp
What are the formulas for the relationships of radians?
Arc Length= (radius)(number of radians)
Radius= arc length/ nubmer of radians
Number of Radians= arc length/ radius
One Complete Revolution= 2pi radians
Angular Velocity= number of radians/ time
Values of Trigonometric ratios in: Quadrant 2
Sin is positive and Cos and Tan are negative. The value of theta is 1800- theta.
Identify the Reciprocal Identities.
csc|theta|=1/sin|theta|
sec|theta|=1/cos|theta|
cot|theta|=1/tan|theta|=cos|theta|/sin|theta|
Describe the Mapping Rule.
(x,y) -> (x/k+d,a(y)+c)
In the function f(x)=sin(k(x)), what affect does k have on the function?
*horizontal reflection(reflection in the y-axis) if k<0
*horizontal compression if k>1 or k<-1
*horizontal stretch if -1<k<1
-this affects the period
What are the formulas for the relationship between radians and degrees conversion?
Radians to Degrees:
2pi radians=3600
1 Radian= (1800/pi)
Degrees to Radians:
3600= 2pi radians
10= (pi/ 180)rad
Values of Trigonometric ratios in: Quadrant 3
Tan is positive and Sin and Cos are negative. The value of theta is 1800+ theta.
Identify the Opposite Angle Identities.
sin(-theta)=-sin(theta)
cos(-theta)=cos(theta)
tan(-theta)=-tan(theta)
What is the formula for amplitude?
a=max-min/2
In the function f(x)=sin(x-d), what affect does d have on the function?
*horizontal translation(phase shift) right if d>0
*horizontal translation left if d<0
-this does not affect the period, amp, or axis of the curve
Convert the values of the following Degrees into Radians:
300
450
600
900
300= pi/6
450= pi/4
600= pi/3
900= pi/2
Values of Trigonometric ratios in: Quadrant 4
Cos is positive and Sin and Tan are negative. The value of theta is 3600- |theta|.
Identify the Double Angle Identities.
sin2(theta)=2sin(theta)cos(theta) =2tan(theta)/1+tan2(theta)
cos2(theta)=cos2(theta)-sin2(theta) =2cos2(theta)-1 =1-2sin2(theta) =1-tan2(theta)/1+tan2(theta)
tan2(theta)=2tan(theta)/1-tan2(theta)
What is the formula for the Equation of the Axis of the Curve?
y=max+min/2
In the function f(x)=sin(x)+c, what affect does c have on the function?
*vertical translation up if c>0
*vertical translation down if c<0
-this affects the axis of the curve
What is the difference between an approximate answer and an exact answer?
An exact answer is an answer where the numbers are not reduced or rounded. An approximate answer consists of the number that is rounded or a decimal.
For the Unit Circle, what is the equivalent to (x,y)?
The equivalent ordered pair to (x,y) in relation to the Unit Circle is (cos|theta|, sin|theta|).
Identify the Compound Angle Formulas.
sin(x+y)=(sin(x))(cos(y))+(cos(x))(sin(y))
sin(x-y)=(sin(x))(cos(y))-(cos(x))(sin(y))
cos(x+y)=(cos(x))(cos(y))-(sin(x))(sin(y))
cos(x-y)=(cos(x))(cos(y))+(sin(x))(sin(y))
tan(x+y)=(tan(x))+(tan(y))/1-(tan(x))(tan(y))
tan(x-y)=(tan(x))-(tan(y))/1+(tan(x))(tan(y))
Is there a relationship between the value of k in the general equation f(x)=(a)sin[k(x-d)]+c, and the number of cycles completed in 2pi?
Yes, k represents the number of cycles in this domain.
Is there a value of d that could make the statement sin(x-d)=cos(x) true?