θ lies in quadrant III. = 12 38. 5 13 3 θ and 3π < < 2π (quadrant IV) cosθ 2 40. Using the Pythagorean Identities: Using the Pythagorean Identities: θ = − θ sin 1 cos2 2 θ θ sin = 1 − cos2 2 2 2 θ sin = 1 − ⎛ − 2 θ 3 ⎞ = 1 − 9 = sin = 1 − ⎛ = 1 − 144 = 25 2 16 12 ⎞ ⎜ ⎟ 5 25 25 169 169 ⎜ 13 ⎟ ⎝ ⎠ ⎝ ⎠ θ sin =± 16 4 25 5 θ =± = sin 25 5 169 13 θ θ Note that sin must be negative because lies θ θ Note that sin must be negative because lies θ in quadrant III. Thus, sin 4 θ in quadrant IV. Thus, sin 5 . 5 =− 13 ⎞ ⎞ 5 4 5 θ sin ⎛ ⎛ 4 5 − θ 5 sin 13 − 4 θ tan = = 5 θ = − ⎜ − ⎟ = tan = = ⎜ ⎟ = − 13 = − cosθ 3 cosθ ⎝ ⎠ ⎝ ⎠ 5 3 3 13 12 12 12 cscθ = 1 = 1 = − cscθ = 1 = 1 = − 13 5 θ sin 4 5 4 sin 13 θ secθ = 1 = 1 = − secθ = 1 = 1 = 13 5 cosθ 3 3 cosθ 12 θ = = = cot 1 1 θ cot = 1 = 1 = − 12 3 θ tan 4 5 12 θ tan θ and 180º < < = 1 (quadrant IV) 39. 270º 3 2 tan and 3π < 5 sin θ =− θ (quadrant III) θ 41. < 2π 13 Using the Pythagorean Identities: Using the Pythagorean Identities: θ θ cos2 = 1 − sin 2 θ θ = + sec2 tan 1 2 2 2 θ = − ⎛ − ⎞ cos2 1 5
38. cosθ 3 θ lies in quadrant III. 5 40. = 12 13 and 3π < θ < 2π (quadrant IV) 2 Using the Pythagorean Identities: sin 2 θ = 1 − cos2 θ 2 Using the Pythagorean Identities: sin 2 θ = 1 − cos2 θ 2 sin 2 θ = 1 − ⎛ − 3 ⎞ = 1 − 9 = 16 sin 2 θ = 1 − ⎛ 12 ⎞ = 1 − 144 = 25 ⎜ 5 ⎟ 25 25 ⎜ 13 ⎟ 169 169 ⎝ ⎠ sin θ =± 16 4 25 5 Note that sin θ must be negative because θ lies in quadrant III. Thus, sin θ 4 5 ⎝ ⎠ sin θ =± 25 = 5 169 13 Note that sin θ must be negative because θ lies in quadrant IV. Thus, sin θ 5 . =− 13 sin θ − 4 4 ⎛ 5 ⎞ 4 sin θ − 5 5 ⎛ 13 ⎞ 5 tan θ = = 5 = − ⎜ − ⎟ = tan θ = = 13 = − ⎜ ⎟ = − cosθ 3 5 ⎝ 3 ⎠ 3 cosθ 12 13 ⎝ 12 ⎠ 12 cscθ = 1 = 1 = − 5 cscθ = 1 = 1 = − 13 sin θ 4 4 sin θ 5 13 secθ = 1 = 1 = − 5 secθ = 1 = 1 = 13 cosθ 3 3 cosθ 12 cot θ = 1 = 1 = 3 cot θ = 1 = 1 = − 12 tan θ 4 tan θ 5 12 39. 5 sin θ =− 13 and 3π < θ < 2π (quadrant IV) 2 41. tan θ = 1 3 and 180º < θ < 270º (quadrant III) Using the Pythagorean Identities: cos2 θ = 1 − sin 2 θ 2 Using the Pythagorean Identities: sec2 θ = tan 2 θ + 1 2 cos2 θ = 1 − ⎛ − 5 ⎞
62514-0132318806_ISM_02-1.docx