CHAPTER 5 IDENTITIES PYTHAGOREAN IDENTITIES Sin 2 Cos
- Slides: 40
CHAPTER 5 IDENTITIES
PYTHAGOREAN IDENTITIES • Sin 2θ+ Cos 2θ = 1 • Tan 2θ+ 1 = Sec 2θ • 1+ Cot 2θ= Csc 2θ Cos 2θ = 1 - Sin 2θ = 1 - Cos 2θ 1 = Sec 2θ- Tan 2θ = Sec 2θ- 1 Cot 2θ = Csc 2θ- 1 1 = Csc 2θ - Cot 2θ
Quotient Identities
Reciprocal Identities
Most popular 2 Sin θ+ 2 Cos θ =1
Pythagorean Identities • Use the pythagorean identities to find cosθ. Given each of the following: a) tanθ= -5/3 in QII b) sinθ= 2/7 in QI c) cscθ= -9/4 in QIII
Pythagorean Identities
Write in terms of cosθ Write tanθsinθ in terms of cosθ Start DONE
Write in terms of cosθ Write cot 2θsin 2θ+ secθ in terms of cosθ Start DONE
Write in terms of cosine • Practice the following a) secxcotxsinx
Verify the identity Show that they are equal. You must clearly show each step. cotx + 1 = cscx(cosx + sinx) Choose the left side or the right side to work on. = cscxcosx + cscxsinx = cotx + 1 DONE
Verify the identity tan 2 Asin 2 A = tan 2 A + cos 2 A - 1 tan 2 A(1 -cos 2 A) = tan 2 A – tan 2 Acos 2 A = tan 2 A – sin 2 A = tan 2 A –(1 - cos 2 A) = tan 2 A – 1 +cos 2 A = DONE
Verify the identity DONE
Verify the identity • PRACTICE THE FOLLOWING a. cos 2 x(tan 2 x + 1) = 1 b. c. sin 2 xsec 2 x + sin 2 xcsc 2 x = sec 2 x
Verify the identity
Sum & Difference Identitites sin(A + B) = sin. Acos. B + cos. Asin. B sin(A – B) = sin. Acos. B – cos. Asin. B cos(A + B) = cos. Acos. B – sin. Asin. B cos(A – B) = cos. Acos. B + sin. Asin. B
Find the exact value Sin 75° Think of two degree measurements that are commonly used on the unit circle that add up to 75. sin(A + B) = sin. Acos. B + cos. Asin. B sin(30° + 45°) = sin 30°cos 45° + cos 30°sin 45°
Find the exact value cos(-5π/12) This problem you need to use the difference identity for the cosine. cos(A - B) = cos. Acos. B + sin. Asin. B
Find the exact value • Practice the Following a) cos(-15°) b) sin(π/12) c) cos(7π/12)
Sum & difference • Find the exact value This is the expanded form of one of the sum and difference identi cos(A - B) = cos. Acos. B + sin. Asin. B A = 120 and B = 30 0
Sum & difference • Practice
Sum & difference verifying identities sin. Acos. B + cos. Asin. B cos. Acos. B - sin. Asin. B DONE
Sum & difference verifying identities • PRACTICE a) sin(x + y) + sin(x – y) = 2 sinxcosy b) cos(300° - x) + sin(30° + x) = cosx
Evaluate • If sin. A = ⅘ and cos. B = -5/13, where A is in quadrant II and B is in quadrant III. Find sin(A + B) = sin. Acos. B + cos. Asin. B
Practice a) cos x = 2/3 and sin y = -1/3. x is in quadrant II and y is in quadrant IV. Find sin(x – y) & sin(x + y)
Sum & Difference Identities
Practice • Use the Tangent sum and difference for the following: a) tan 105° b) tan 285° c) tan(-13π/12)
Practice • Use the Tangent sum and difference for the following: Given cos. A = 3/5 and sin. B = 5/13 A&B in QI Find tan(A+B)…. Using the pythagorean and finding x, y, & r we get tan. A = 4/5 and tan. B = 5/12
Double-Angle Identities
Practice • Given cosθ=3/5 and sin<0 Find sin 2θ and cos 2θ “use Double-Angle Identities”
Practice • Find all 6 trig function values given: cos 2θ=4/5 θ in QII “use Double-Angle Identities”
Half-Angle Identities The ± is determined by the Quadrant of the original angle
Practice • Find the exact value(use the Half-Angle Identities) a) cos 15° b) sin 165°
- Nilai dari 8 sin 112,5° cos 22,5° =
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