Mod E - Trigonometry · As a point moves along a circle or radius r, its angular velocity, ω= θ_ ω, is the angle or rotation per unit of time; its t linear velocity, v, is the

E.0. CirclesE.1. AnglesE.2. Right Triangle TrigonometryE.3. Points on Circles Using Sine and CosineE.4. The Other Trigonometric FunctionsE.5. Sinusoidal GraphsE.6. Graphs of the Other Trigonometric Functions

E.0. CirclesRemember the standard form of a circle: (x-h)2 + (y-k)2 = r2

The equation of a circle with radius 1 centered at the origin is: ______________

E.1. AnglesAn angle is the area of a plane between two rays with common endpoint. One of the rays is called the initial side and the other is called the terminal side of the angle.

An angle in standard position has its vertex at the origin and its initial side on the positive x-axis. Theterminal side may be in any of the four quadrantsor on any of the axes. We use Greek, lower-case letters to indicate angles: α (alpha), β (beta), θ (theta), φ orϕ (phi)Angles that open counterclockwise are positive.Angles that open clockwise are negative. Angles may be measured in degrees. There are360⁰ in one complete revolution (circle).This means that 1/4 circle is __________ 1/4 circle is _________ 3/4 of a circle are __________ right angle straight angle

An angle with terminal side in the first quadrant is _______________ acute

An angle with terminal side in the second quadrant is ____________ obtuse

An angle with terminal side in the third quadrant is ______________ reflex

An angle with terminal side in the fourth quadrant is _____________ reflex

Mod E - TrigonometryWednesday, July 27, 2016 12:13 PM

M132-Blank NotesMOM Page 1

Practice drawing angles of 30⁰, 45⁰, and 60⁰Draw angles of 120⁰, 135⁰, and 150⁰Draw the angle with measure -300⁰

Two angles in standard position are called coterminal if their terminal sides coincide (fall on each other).The difference of their measures is a multiple of ________If the one angle is 390⁰, what is the measure of the acute, coterminal angle? ____________What is a negative angle coterminal with both? _________

Find two angles that are coterminal with 135⁰. __________________________Find an angle of least positive measure (0⁰≤θ<360⁰) coterminal with

1070⁰ _______________________-65⁰ ________________________

What is the expression for all angles coterminal with 90⁰? __________________

How many degrees of longitude arebetween two different time zones on earth?

Radian MeasureArclength is the length of an arc, s, along a circle of radius r, subtended (drawn out) by and angle θ.One radian is the measure of the angle that subtends an arc of length r.

For θ measured in radians, s = r*θ


How many radians is a complete circle?

____________________________Therefore, 360⁰ = 2π, and 180⁰ = πConvert to radians:

90⁰ = _______ 30⁰ = ________ 60⁰ = _________

45⁰ = _______ 120⁰ = ________ 150⁰ = _________ 270⁰ = ________

-135⁰ = ________ -20⁰ = _________ -210⁰ = _________ -240⁰ = _________

If the radius of the earth is 3960 miles, what is the distance from the equator of a point at 40⁰N latitude?

A circle has radius 9.5 cm. Find the length ofthe arc intercepted by a 120⁰ central angle.

Convert from radians to degrees:π/6 = ___________ 4π/3 = ___________ -3π/4 = ___________

Find angles 0≤θ<2π coterminal with

-17π/6 _________________29π/3 __________________-18π/5 _________________

Area of a Sector: To find the area of a sector πr2 = A? ==> A = of a circle with radius r 2π θSubtended by an angle θ, OR πr2 = A? ==> A =solve the proportion for A: 360⁰ θ


In center pivot irrigation, a large irrigation pipe on wheels rotates around a center point. If the irrigation pipe is 450m long, how mucharea can be irrigated after a rotation of 240⁰?

Angular and Linear VelocityAs a point moves along a circle or radius r, its angular velocity, ω = θ_ω, is the angle or rotation per unit of time; its tlinear velocity, v, is the distance travel per unit of time. v = s_ tSince s = r*θ, v = s / t = (r*θ) / t = r*(θ/t) = r*ω ==> v = r*ω

A tire with radius 9 inches is spinning at 80 rev/min.Find the angular speed of the tire in radians/min _________________________

Find the speed in inches/min and miles/h _______________________________

_________________________________________________________________

E.2. Right Triangle Trigonometry Definitions (SOH CAH TOA) sinθ = opp cscθ = hyp hypotenuse hyp opp opposite cosθ = adj secθ = hyp θ hyp adj adjacent tanθ = opp cotθ = adj_ adj opp

Determine all trigonometric functions of angle A.


Determine all trigonometric functions of angle B.

Solve the right triangle given c=12 and A=40⁰

Solve the right triangle given b=8 and B=38⁰

A 12ft ladder leans against a building so the angle between the ladder and the ground is 72⁰. How high will the ladder reach? Round to the nearest tenth.

A radio tower is located 350 feet from a building. From a window in the


A radio tower is located 350 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 29⁰, and the angle of depression to the bottom of the tower is 20⁰. How tall is the tower? (Round to the nearest tenth of a foot.)

Find x correct to two decimal places.

Trigonometric Functions of 45⁰, 30⁰, 60⁰

sin45⁰ = cos45⁰ = tan45⁰ = 1 csc(π/4) = sec(π/4) = cot(π/4) = 45⁰ 1

sin30⁰ = cos30⁰ = tan30⁰ =

2 2 csc(π/6) = sec(π/6) = cot(π/6) =

sin60⁰ = cos60⁰ = tan60⁰ =

csc(π/3) = sec(π/3) = cot(π/3) =


E.3. Trigonometric Functions of Any Angle & The Unit CircleFor any angle in standard position sinθ = y cosθ = x tanθ = y r r x

cscθ = r secθ = r scotθ = x y x y

***NOTE: Pay attention to the signs of x,yin the different quadrants. r is always positive (All Students Take Calculus)

The terminal side of an angle in standard positionpasses through the point (8,-6). Find all trig-funs.

***Find all trig-funs of θ if tanθ = -2/3 and cosθ>0.

*** Given sinθ = -1/3 and cosθ < 0, find all trig-funs of θ.

Reference Angle: The acute, positive angle between the terminal side of θ and


Reference Angle: The acute, positive angle between the terminal side of θ and the x-axis. Any angle θ and its reference angle have identical trigonometric functions (except maybe for the signs).

What is the referenceangle for 150⁰? __________Compare their trig-funs.

What is the referenceangle for 4π/3? __________Compare their trig-funs.

What is the referenceangle for -20π/3? _________Compare.

Trigonometric Functions of 0⁰, 90⁰, 180⁰, 270⁰

sin0= sin(π/2)= sin(π)= sin(3π/2)=cos0= cos(π/2)= cos(π)= cos(3π/2)=tan0= tan(π/2)= tan(π)= tan(3π/2)=csc0= csc(π/2)= csc(π)= csc(3π/2)=sec0= sec(π/2)= sec(π)= sec(3π/2)=cot0= cot(π/2)= cot(π)= cot(3π/2)=

Unit Circle: Circle with r=1 and center at the origin: Equation: _______________

sinθ= cosθ= tanθ=

cscθ= secθ= cotθ=

The Tangent and Cotangent Axes M132-Blank NotesMOM Page 8

The Tangent and Cotangent Axes

E.4. (More on) The Other Trigonometric FunctionsBasic Trigonometric Identities Pythagorean identity: cos2θ + sin2θ = 1

Ratio Identities: tanθ = sinθ cotθ = cosθ cosθ sinθ

Reciprocal Identities: cscθ = 1 _ secθ = 1 _ cotθ = 1 _ sinθ cosθ tanθ

Cofunction Identities: sin(90⁰-θ) = cos(θ) cos(90⁰-θ) = sin(θ) tan(90⁰-θ) = cot(θ) cot(90⁰-θ) = tan(θ) sec(π/2-θ) = csc(θ) csc(π/2-θ) = sec(θ)

Even/Odd Identities: cos(-θ) = cos(θ) sec(-θ) = sec(θ) (even) sin(-θ) = -sin(θ) csc(-θ) = -csc(θ) (odd) tan(-θ) = -tan(θ) cot(-θ) = -cot(θ) (odd)

Prove the other forms of the Pythagorean Identity: 1+tan2θ = sec2θ cot2θ + 1 = csc2θ

***NOT in MOM: If sinθ = a, write an algebraic expression for cosθ = _______________________

If sinθ = a, cosθ = b, and tanθ = c, write an algebraic expression for cscθ + cos(π/2-θ) + tan(-θ) = ___________________

If sinθ = a, cosθ = b, and tanθ = c, write an M132-Blank NotesMOM Page 9

If sinθ = a, cosθ = b, and tanθ = c, write an algebraic expr. for sin(4π+θ) - tan(π/2-θ) + cos(-θ) = ____________________

E.5. Sinusoidal GraphsThe Graph of the Sine Function

Domain: _____________ Range: _____________

Period: ______________ Amplitude: __________

Even/Odd? __________

The Graph of the Cosine Function

Domain: _____________ Range: _____________

Period: ______________ Amplitude: __________

Even/Odd? __________

Variations of the Sine and Cosine Functions y= A sin(Bx-C) + D


Variations of the Sine and Cosine Functions y= A sin(Bx-C) + DAmplitude: |A| Phase Shift: C/B Period (P): 2π/B Midline: y=D5 Points with x coordinates: C/B, (C/B+P/4), (C/B+2P/4), (C/B+3P/4), (C/B+4P/4)***NOTE: If A>0, we graph the usual sine/cosine curve If A<0, we graph the curve up-side down (reflected about x-axis)

In MOM: y = Asin(B(x-C)) + DAmplitude: |A| Horiz. Shift: C Period (P): 2π/B Vert. Shift: D5 Points with x coordinates: C, (C+P/4), (C+2P/4), (C+3P/4), (C+4P/4)

Graph one cycle of 2 sin(2 (x + π/4)) - 1

Amplitude: ______ Phase Shift: ______ Period: ______ Midline: _____

Coordinates of 5 points: ___________________________________________

Graph one cycle of f(x) = 1/2 cos(1/2 x - π/4) + 2



Graph one cycle of f(x) = 4 - sin(π (x+1))


Graph one cycle of f(x) = 4 - sin(π (x+1))



Determining the Equation from the Graph

Midline: _________ ==> D = ___________

Period: __________ ==> B = ___________

Phase Shift: ______ ==> C = ___________

Amplitude: ______ Direction of Graph: ______

==> A = ________ Equation: y = ____________________

E.6. Graphs of Other Trigonometric FunctionsWatch the videos just to have an idea of how the graphs look like. Not tested or covered in this class. No homework for this section.


Mod E - Trigonometry · As a point moves along a circle or radius r, its angular velocity, ω= θ_ ω, is the angle or rotation per unit of time; its t linear velocity, v, is the

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