θ 90º < < 180º θ − 4 and π< < 3π 42. 3 2 2 44. cscθ = θ tan (quadrant II) (quadrant III) Using the Pythagorean Identities: Using the Pythagorean Identities: θ = θ + sec2 tan 1 2 θ θ cot = csc2 − 1 2 θ = ⎛ − sec2 2 + 1 = 4 + 1 = ⎞ (−4)2 − 1 = 16 −1 = 15 cot θ = 13 2 ⎜ ⎟ 3 9 9 ⎝ ⎠ θ cot =± 15 13 secθ =± =± θ θ Note that cot must be positive since lies in 9 3 θ = quadrant III. Thus, cot 15 . θ Note that secθ must be negative since lies in θ = = = − sin 1 1 1 quadrant II. Thus, secθ = − 13 . −4 cscθ 4 3 cosθ = 1 = 1 = − 3 ⋅ 13 = − 3 θ = cot cosθ , so θ sin 13 secθ (cot ) (sin θ θ cosθ = 15 ⎛ − 1 ⎞ = − 15 − 13 13 13 13 ) = 3 ⎜ 4 ⎟ 4 ⎝ ⎠ θ tan = sin θ = = ⋅ = , so tan 1 1 15 15 θ cosθ 15 15 15 θ cot ( ) (cosθ ) θ θ = = − ⎛ − ⎞ = sin tan 2 3 13 1 1 4 15 4 15 2 13 ⎜ ⎟ secθ = = = cosθ − − = − ⋅ 15 15 15 3 ⎝ 13 ⎠ 13 15 π cscθ = 1 = 1 = 13 = 13 θ sin < θ < π θ 45. cot = − 2 2 13 2 13 2 (quadrant II) and 13 2 θ cot = 1 = 1 = − Using the Pythagorean Identities: θ θ csc2 = 1 + cot 3 2 2 θ tan 2 Using the Pythagorean 2 Identities: θ θ tan = sec2 −1 θ 43. secθ = 3 and 3π < < 2π (quadrant IV) 2
42. tan θ 2 90º < θ < 180º 3 (quadrant II) 44. cscθ = − 4 and π< θ < 3π 2 (quadrant III) Using the Pythagorean Identities: sec2 θ = tan 2 θ + 1 Using the Pythagorean Identities: cot 2 θ = csc2 θ − 1 sec2 θ = ⎛ − 2 ⎞ + 1 = 4 + 1 = 13 cot 2 θ = (−4)2 − 1 = 16 −1 = 15 ⎜ 3 ⎟ 9 9 ⎝ ⎠ 13 secθ =± =± cot θ =± 15 Note that cot θ must be positive since θ lies in 9 3 Note that secθ must be negative since θ lies in quadrant III. Thus, cot θ = sin θ = 1 = 1 = − 1 15 . quadrant II. Thus, secθ = − 13 . cscθ −4 4 3 cosθ = 1 = 1 = − 3 ⋅ 13 = − 3 13 cot θ = cosθ , so sin θ secθ − 13 13 13 13 cosθ = (cot θ ) (sin θ ) = 15 ⎛ − 1 ⎞ = − 15 3 ⎜ 4 ⎟ 4 tan θ = sin θ , so ⎝ ⎠ tan θ = 1 = 1 ⋅ 15 = 15 cosθ cot θ 15 15 15 sin θ = ( tan θ ) (cosθ ) = − 2 ⎛ − 3 13 ⎞ = 2 13 1 1 4 15 4 15 ⎜ ⎟ secθ = = cosθ − = − ⋅ 15 = − 15 15 3 ⎝ 13 ⎠ 13 15 cscθ = 1 = 1 = 13 = 13 π sin θ 2 13 2 13 2 13 45. cot θ = − 2 and < θ < π (quadrant II) 2 cot θ = 1 = 1 = − 3 Using the Pythagorean Identities: csc2 θ = 1 + cot 2 θ tan θ 2 2 43. secθ = 3 and 3π < θ < 2π (quadrant IV) 2 Using the Pythagorean Identities: tan 2 θ = sec2 θ −1
62514-0132318806_ISM_02-1.docx