CHAPTER 5 IDENTITIES PYTHAGOREAN IDENTITIES Sin 2 Cos

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CHAPTER 5 IDENTITIES

CHAPTER 5 IDENTITIES

PYTHAGOREAN IDENTITIES • Sin 2θ+ Cos 2θ = 1 • Tan 2θ+ 1 =

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

Quotient Identities

Reciprocal Identities

Reciprocal Identities

Most popular 2 Sin θ+ 2 Cos θ =1

Most popular 2 Sin θ+ 2 Cos θ =1

Pythagorean Identities • Use the pythagorean identities to find cosθ. Given each of the

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

Pythagorean Identities

Write in terms of cosθ Write tanθsinθ in terms of cosθ Start DONE

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

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

Write in terms of cosine • Practice the following a) secxcotxsinx

Verify the identity Show that they are equal. You must clearly show each step.

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

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 DONE

Verify the identity • PRACTICE THE FOLLOWING a. cos 2 x(tan 2 x +

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

Verify the identity

Sum & Difference Identitites sin(A + B) = sin. Acos. B + cos. Asin.

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

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

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)

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

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 • Practice

Sum & difference verifying identities sin. Acos. B + cos. Asin. B cos. Acos.

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 –

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

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

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

Sum & Difference Identities

Practice • Use the Tangent sum and difference for the following: a) tan 105°

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

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

Double-Angle Identities

Practice • Given cosθ=3/5 and sin<0 Find sin 2θ and cos 2θ “use Double-Angle

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

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

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

Practice • Find the exact value(use the Half-Angle Identities) a) cos 15° b) sin 165°