Sec 4.1 Trigonometric Identities Basic Identities. Name: Reciprocal Identities:
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1 Sec 4. Trigonometric Identities Basic Identities Name: Reciprocal Identities: Quotient Identities: sin csc cos sec csc sin sec cos sin tan cos cos cot sin tan cot cot tan Using the Reciprocal and Quotient Identities simplify each as much as possible.. tan(θ) cos(θ). cot(θ) cos(θ) sin(θ) 3. sin(θ) cot(θ) + cos(θ) 4. csc(θ) tan(θ) sec(θ) 5. cot(θ) csc(θ) 6. cot(θ) sin(θ) cos(θ) + cos(θ) sec(θ) 7. tan (θ) csc(θ) cos(θ) sec(θ) M. Winking (Section 4-) p.56
2 Reciprocal Identities: Quotient Identities: sin csc cos sec tan cot csc sin sec cos cot tan sin tan cos cos cot sin Using the Reciprocal and Quotient Identities verify the following trigonometric identities. 8. cot(θ) sec(θ) sin(θ) = 9. cot(θ) csc(θ) = cos(θ) 0. sin(θ)sec(θ) tan(θ) + cos(θ) sec(θ) =. tan(θ) + sec(θ) = +sin(θ) cos(θ). csc(θ) + sec(θ) = cos(θ)+sin(θ) cos(θ)sin(θ) 3. csc(θ) + tan(θ) cos(θ) = sin(θ) M. Winking (Section 4-) p.57
3 Consider the following diagram. Using basic trigonometry solve for x in terms of. y Using basic trigonometry solve for y in terms of. x Write a Pythagorean Theorem statement using these expressions. Pythagorean Identities: Using the reciprocal identities verify tan sec sin tan cos sec cot csc Using the reciprocal identities verify cot csc Using the Reciprocal, Quotient, and Pythagorean Identities simplify each as much as possible. 4. sin θ+cos θ sin θ 5. sin θ (sin θ + cos θ cot θ) M. Winking (Section 4-) p.58
4 Reciprocal Identities Quotient Identities Pythagorean Identities sin θ = csc θ = sin θ tan θ = sin θ + cos θ = csc θ sin θ cos θ cos θ = sec θ = cos θ cot θ = tan θ + = sec θ sec θ cos θ sin θ tan θ = cot θ cot θ = tan θ + cot θ = csc θ Using the Reciprocal, Quotient, and Pythagorean Identities simplify each as much as possible. 6. sec α sec α sin α 7. sin θcotθ sin θ 8. (sec θ+)(sec θ ) sin θ 9. csc x cos x cot x sin x M. Winking (Section 4-) p.59
5 Reciprocal Identities Quotient Identities Pythagorean Identities sin θ = csc θ = sin θ tan θ = sin θ + cos θ = csc θ sin θ cos θ cos θ = sec θ = cos θ cot θ = tan θ + = sec θ sec θ cos θ sin θ tan θ = cot θ cot θ = tan θ + cot θ = csc θ Using the Reciprocal, Quotient, and Pythagorean Identities verify each trigonometric identity. 0. csc (x) csc (x) cos (x) =. sin θ(cot θ + tan θ) = sec θ. tan θ + cot θ = sec θ csc θ tan sin 3. tan cos M. Winking (Section 4-) p.60
6 Consider that by definition an odd function is symmetric about the origin (fold along the x-axis and the y-axis): Therefore, if a function is odd we also know that f( x) = f(x) which would suggest sin( x) = sin(x) Consider that by definition an even function is symmetric about the y-axis (fold along the y-axis): Therefore, if a function is even we also know that f( x) = f(x) which would suggest cos( x) = cos(x) Even-Odd Identities: sin( x) = sin(x) cos( x) = cos(x) tan( x) = tan(x) 4. Verify: sec(x) = sin (x) + cos (x) cos( x) 5. Verify: csc(x) = sin (x) sin( x)
7 Sec 4. Trigonometric Identities Sum & Difference Identities Name: Consider the diagram at the right of a unit circle.. First, determine the coordinates of point A in terms of α.. First, determine the coordinates of point B in terms of β. 3. Using those coordinates and the distance formula, find the distance between AB in terms of α and β. 4. Using the Law of Cosines and triangle ABC, find the length of side AB in terms of α and β. 5. Use the two unique descriptions of the length of AB create a new trigonometric identitity. M. Winking (Section 4-) p.6
8 Alternately, you could use the following elegant diagram to show the sum identities hold true for angles from 0 to 90 or 0 to π by realizing opposite sides of a rectangle must have the same measure. Further, the difference identities can be determined by replacing β with negative β and simplifying. sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β sin 75. Find the exact value of (using sum and difference identities) sin sin cos cos sin. Find the exact value of cos 55 (using sum and difference identities) cos cos cos sin sin M. Winking (Section 4-) p.6
9 3. Find the exact value of (using sum and difference identities) sin 3 sin sin cos cos sin 4. Find the exact value of (using sum and difference identities) cos 5 cos cos cos sin sin Given that sina and cosb sin cos cos sin Find the exact value of sin A B = sin 3. Also, assume A & B are in the first quadrant. 5 M. Winking (Section 4-) p.63
10 Given that cosa and cosb cos cos sin sin Find the exact value of cos A B= cos. Also, assume A & B are in the first quadrant. 5 Simplify the following trigonometric expressions using the sum and difference identities. 7. sin(π + θ) 8. cos(θ + θ) M. Winking (Section 4-) p.64
11 Determine the sum identity for tangent using the sum identities for sine and cosine. Tan(α + β) = sin(α+β) sin α cos β+cos α sin β = cos(α+β) cos α cos β cos α cos β cos α cos β sin α sin β cos α cos β cos α cos β = tan Find the exact value of (using sum and difference identities) tan tan tan tan tan 5 3 tan tan tan tan tan 0. Given that sina and cosb Find the exact value of tan A B= 8 7. Also, assume A & B are in the first quadrant. M. Winking (Section 4-) p.65
12 Sec 4.3 Trigonometric Identities Double Angle Identities Name: Starting with the sum and difference identities, create the double angle identities: sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β. Simplify sin(θ + θ). Simplify cos(θ + θ) 3. Using the Pythagorean Identities, find new ways to write the double angle formula for cosine. 4. Simplify tan(θ + θ) using the sum identity tan(α + β) = tan α+tan β tan α tan β M. Winking (Section 4-3) p.66
13 Double-Angle Identities cos(θ) = cos θ sin θ sin(θ) = sin θ cos θ cos(θ) = sin θ tan(θ) = cos(θ) = cos θ 5. Given that sin a. sin(θ) 5 and angle A lies in the first quadrant, find the exact value of each of the following: 3 tan θ tan θ b. cos(θ) c. tan(θ) 6. Given that cos a. sin(θ) 5 and angle A lies in the first quadrant, find the exact value of each of the following: b. cos(θ) c. tan(θ) M. Winking (Section 4-3) p.67
14 Simplify the following trigonometric expressions using the sum and difference identities. 7. cos θ sin θ = sin(θ) 8. sin(x) + = (sin x + cos x) 9. cos(x) = (cos x + sin x)(cos x sin x) 0. cos θ = cot θ sin θ M. Winking (Section 4-3) p.68
15 Sec 4.3 Trigonometric Identities Half Angle Identities Starting with the double angle identities, create the half angle identities: Name: Double-Angle Identities cos(θ) = cos θ sin θ sin(θ) = sin θ cos θ cos(θ) = sin θ tan(θ) = cos(θ) = cos θ tan θ tan θ. Let θ = α using the identity cos(θ) = sin θ and solve for sin θ. Let θ = α using the identity cos(θ) = cos θ and solve for cos θ 3. Use the two identities you just created in problems and, simplify the following to create the tangent half angle identity. tan θ = sin (θ ) cos ( θ ) = M. Winking (Section 4-4) p.69
16 Half-Angle Identities sin ( θ θ ) = ± cos cos ( θ θ ) = ± +cos tan ( θ θ ) = ± cos +cos θ 4. Find the exact value of cos.5 (using half angle identities) 5. Find the exact value of sin (using half angle identities) 6. Find the exact value of tan 5 (using half angle identities) M. Winking (Section 4-4) p.70
17 5 and assume angle is in the first quadrant. 3 a. Find the exact value of sin = 7. Given that sin b. Find the exact value of cos = c. Find the exact value of tan = 8. Verify the following identities using the half angle identities. a. cos ( θ ) = cos θ b. sin θ +cos θ = tan θ M. Winking (Section 4-4) p.7
18 Sec 4.5 Trigonometric Identities Trigonometric Equations & Applications Name: Find the general solution to the following in degrees using a roster solution and an algebraic solution.. sin(x) = 3 y = 3 Roster Solution: Algebraic Solution: Find the general solution to the following in radians using a roster solution and an algebraic solution.. cos(x) = y = Roster Solution: Algebraic Solution: M. Winking (Section 4-5)
19 Find the general solution to the following in degrees using a roster solution and an algebraic solution. 3. cos(x) + 4 = 3 Roster Solution: Algebraic Solution: Find all solutions to the following problem in radians using a roster solution and an algebraic solution. 4. sin(x) = 3 Roster Solution: Algebraic Solution: M. Winking (Section 4-5) p.73
20 Find all solutions to the following problems in radians using a roster solution and an algebraic solution. 5. sin(x) sin(x) = 0 Roster Solution: Algebraic Solution: 6. cos(x) cos(x) = 0 Roster Solution: Algebraic Solution: M. Winking (Section 4-5) p.74
21 Using your graphing calculator solve the following. 7. The average high temperature of a day in Atlanta can be modeled by the following equation: T = 0sin(0.07(d +.86)) + 69 Where T represents the temperature in Fahrenheit and d is the day number of year (e.g. Feb. nd would be day 33) a. Using the model, what is the average high temperature in Atlanta on February th? b. Using the model, what dates should the average high temperature be 8 F? 8. The height (in feet) above aground of a child in a swing can be given by the function with respect to time in seconds. (starting from the swings highest point) h = 3sin ( π 3 (t )) + 4 a. Using the model, what is the height of the child after seconds? b. Using the model, at what times is the height of the child 6 feet high?
22 Trigonometric Identities Reciprocal Identities: sin csc cos sec tan cot csc sin sec cos cot tan Sum & Difference Identities: sin cos sin cos cos sin cos cos sin sin tan tan tan tan tan Quotient Identities: sin cos tan cot cos sin Pythagorean Identities: sin tan cos sec cot csc Double-Angle Identities: sin cos sin cos cos tan sin sin cos tan tan Half-Angle Identities: Co-function Identities: sin cos90 cos sin90 sin cos tan cot90 cot tan90 sec csc90 csc sec90 Opposite-Angle Identities: sin cos A sin A A cos A cos cos cos tan, cos cos De Moivre s Theorem: n cosx sin x cosnxsinnx
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