Identities and Simplifying
Reciprocal Identities... (there should be six)
let x = theta
csc x = (1/sin x)
sec x = (1/cos x)
cot x = (1/tan x)
sin x = (1/csc x)
cos x = (1/sec x)
tan x = (1/cot x)
Describe the trends of the following parent functions:
1) Linear
2) Quadratic
3) Absolute Value
4) Cubic
1) The equation of a linear function is y = x. It correlates a direct relationship between the x and y axis.
2) A Quadratic parent function can be represented by the equation y = x2. The end behavior of the parent function, f(x) = x2, can be described, as x reaches infinity, f(x) reaches infinity, and as x reaches negative infinity, f(x) reaches infinity.
3) Represented by the equation, y = lxl, the parent function is symmetric about the y axis beginning at the origin, (0,0).
4) y = x3 the end behavior of the parent function, f(x) = x3 can be represented by, as x approaches infinity, f(x) approaches infinity, and as x approaches negative infinity, f(x) approaches negative infinity.
Equation for mid-line...
Mid-Line equals the maximum plus the minimum divided by 2
The measurements of the sides of the 90-30-60 degree triangle
hypotenuse is 1, side opposite of 30 degrees in 1/2 and side opposite of 60 degrees is the square route of 3 over 2.
Draw and label a unit circle.
Should be a circle with the four points in this order counterclockwise.
(1,0)
(0,1)
(-1, 0)
(0, -1)
Tangent and Cotangent Identities... (there should be two)
let x = theta
tan x = (sin x/cos x)
cot x = (cos x/sin x)
Describe the parent functions of the sine and cosine graphs. (include, starting point, amplitude, period, and mid-line.)
Sine:
starting point: mid-line
amplitude: 1
period: 2π
mid-line: origin or y = 0
cosine:
starting point: maximum
amplitude: 1
period: 2π
mid-line: y = 0
Equation for Amplitude....
Amplitude equals the maximum minus the minimum divided by two
the hypotenuse is 1, and both the sides across from the 45 degree angles are the square route of 2 over 2.
The vertices of triangle ABC have coordinates A(0,0), B(12,5), and C(12,0).
1) argue that triangle ABC is a right triangle
b) what are the coordinates where the hypotenuse of triangle ABC intersects the unit circle x2 + y2 = 1
c) let theta denote the number of degrees of rotation from side AC to side AB. Calculate sin(0 degrees) and cos(0 degrees)
a) since side AC is a horizontal line and side BC is a vertical line we know that side BC and side AC are perpendicular. Therefore there is a right angle and thus a right triangle.
b) (12/13 , 5/13)
c) sin(0 degrees) = (5/13)
cos(o degrees) = (12/13)
The three Pythagorean identities are... (there should be three)
sin2x + cos2x = 1
1+ cot2x = csc2x
tan2x +1 = sec2x
Identify the amplitude, frequency, period, phase shift, vertical shift, and mid-line. Then write the equation.
(In Problem Packet: Graphing and Transformations: question for 300)
Amplitude: 2
Frequency: 3
Period: (2π/3)
Phase Shift: (π /6) left
Vertical Shift: 1
Mid-line: y = 1
equation: y = 2sin3(x+ (π /6) + 1
Equation for Period and Frequency...
Frequency = b
Period = (2π)/b
Frequency = (2π)/period
(In Problem Packet: Special Right Triangles: question for 300)
Given the diagram as labeled. Find X
6 cm
(In Problem Packet: Circles: question for 300)
use the diagram to explain why sin(135) equals sin(45) but cos(135) does not equal cos(45) - all numbers in terms of degrees
Sin 135 = sin 45 because the opposite sides in both rations is positive. In cos 135 the adjacent side is negative, but in cos 45 it is positive, which makes the values opposite.
Simplify the following....
(In Problem Packet: identities and simplifying: question for 400)
cos x - sin x
Identify the amplitude, frequency, period, phase shift, vertical shift, and mid-line. Then write the equation.
(In Problem Packet: Graphing and Transformations: question for 400)
Amplitude: 2
Frequency: 3
Period: (2π/3)
Phase Shift: none
Vertical Shift: 3
mid-line: y=3
equation: y = 2sin3x+3
1) Conversion formula for converting Radians to degrees, explain format of formula.
2) Conversion formula for converting degrees to radians, explain format of formula.
3) Explain how do you know if something is in radians or degrees.
1) radian value times 180/π. Canceling out pi, since radians in term of pi.
2) degree value times π/180. Canceling out 180 which is in degrees to leave value in terms of pi, radians.
3) no degree symbol, radians, degree symbol in degrees
Martin walks his dog on level ground in a straight line with the dog's favorite tree. The angle of elevation from Martin's present location to the top of a nearby telephone pole is 30 degrees. The angle of elevation from the base of the tree to the top of the telephone pole is 45 degrees. If the telephone pole is 42 feet tall, how far are martin and his dog from the tree. Express the exact length.
(In Problem Packet: Special Right Triangles: question for 400)
42 times radical of 3 minus 42.
Simplify (tan^2 theta - sin^2 theta)/sin^2 theta
tan^2 theta
Simplify the following....
(In Problem Packet: identities and simplifying: question for 500)
2
Identify the amplitude, frequency, period, phase shift, vertical shift, and mid-line. Then write the equation.
(In Problem Packet: Graphing and Transformations: question for 500)
amplitude: 3
Frequency: 1
Period: 2π
Phase shift: none
vertical shift: 1
mid-line: y=1
equation: y = 3sinx + 1
Formula for arc length and the stipulations.
S = r(x)
let x = theta
theta has to be measured in radians
(In Problem Packet: Special Right Triangles: question for 500)
Given the diagram, find a, b, c, and d
DOUBLE THE POINTS IF ALL CORRECT - First team to have all correct get the double points.
a = 16
b = 8
c = 16 times radical 3
d = 24