Mod E - Trigonometry. Wednesday, July 27, M132-Blank NotesMOM Page 1

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1 M132-Blank NotesMOM Page 1 Mod E - Trigonometry Wednesday, July 27, :13 PM E.0. Circles E.1. Angles E.2. Right Triangle Trigonometry E.3. Points on Circles Using Sine and Cosine E.4. The Other Trigonometric Functions E.5. Sinusoidal Graphs E.6. Graphs of the Other Trigonometric Functions E.0. Circles Remember the standard form of a circle: (x-h) 2 + (y-k) 2 = r 2 The equation of a circle with radius 1 centered at the origin is: E.1. Angles An 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. The terminal side may be in any of the four quadrants or 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 are 360⁰ 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

2 M132-Blank NotesMOM Page 2 Practice drawing angles of 30⁰, 45⁰, and 60⁰ Draw angles of 120⁰, 135⁰, and 150⁰ Draw 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 are between two different time zones on earth? Radian Measure Arclength 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

3 M132-Blank NotesMOM Page 3 How many radians is a complete circle? Therefore, 360⁰ = 2π, and 180⁰ = π Convert 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 of the 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 πr 2 = A? ==> A = of a circle with radius r 2π θ Subtended by an angle θ, OR πr 2 = A? ==> A = solve the proportion for A: 360⁰ θ

4 M132-Blank NotesMOM Page 4 In center pivot irrigation, a large irrigation pipe on wheels rotates around a center point. If the irrigation pipe is 450m long, how much area can be irrigated after a rotation of 240⁰? Angular and Linear Velocity 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 distance travel per unit of time. v = s_ t Since 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.

5 M132-Blank NotesMOM Page 5 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.

6 M132-Blank NotesMOM Page 6 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, 60⁰ 45⁰ 1 1 sin45⁰ = cos45⁰ = tan45⁰ = csc(π/4) = sec(π/4) = cot(π/4) = 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) =

7 M132-Blank NotesMOM Page 7 E.3. Trigonometric Functions of Any Angle & The Unit Circle For 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,y in the different quadrants. r is always positive (All Students Take Calculus) The terminal side of an angle in standard position passes 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.

8 M132-Blank NotesMOM Page 8 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 reference angle for 150⁰? Compare their trig-funs. What is the reference angle for 4π/3? Compare their trig-funs. What is the reference angle 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=

9 M132-Blank NotesMOM Page 9 The Tangent and Cotangent Axes E.4. (More on) The Other Trigonometric Functions Basic Trigonometric Identities Pythagorean identity: cos 2 θ + sin 2 θ = 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+tan 2 θ = sec 2 θ cot 2 θ + 1 = csc 2 θ ***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) =

10 M132-Blank NotesMOM Page 10 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 Graphs The Graph of the Sine Function Domain: Period: Even/Odd? Range: Amplitude: The Graph of the Cosine Function Domain: Period: Even/Odd? Range: Amplitude:

11 M132-Blank NotesMOM Page 11 Variations of the Sine and Cosine Functions y= A sin(bx-c) + D Amplitude: A Phase Shift: C/B Period (P): 2π/B Midline: y=d 5 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)) + D Amplitude: A Horiz. Shift: C Period (P): 2π/B Vert. Shift: D 5 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 Amplitude: Phase Shift: Period: Midline: Coordinates of 5 points:

12 M132-Blank NotesMOM Page 12 Graph one cycle of f(x) = 4 - sin(π (x+1)) Amplitude: Phase Shift: Period: Midline: Coordinates of 5 points: 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 Functions Watch 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.