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Notes 17a Name ________________________ Sum and Difference Formulas (5.2) Sum and Difference Formulas 1. cos (α= + β ) cos α cos β − sin α sin β 2. cos (α= − β ) cos α cos β + sin α sin β 3. sin (α= + β ) sin α cos β + cos α sin β 4. sin (α= − β ) sin α cos β − cos α sin β tan α + tan β 5. tan (α + β ) = 1 − tan α tan β tan α − tan β 6. tan (α − β ) = 1 + tan α tan β Using the Sum or Difference Formulas (Forward): Use the sum or difference formulas to evaluate. π 2π 1) cos (135 − 60 ) 2) tan + 4 3 Using the Sum or Difference Formulas (Backwards): Find the exact value of each expression. π 5π − tan tan 18 36 3) cos 70 cos 40 + sin 70 sin 40 4) π 5π 1 + tan tan 18 36 Finding the Exact Value: Find the exact value of each trigonometric function. 5π sin 6) tan 75 5) 12 Using Sum and Difference: 4 1 Suppose that sin α = for a quadrant I angle α and cos β = for a quadrant I angle β . 2 5 7) cos (α + β ) 8) tan(α + β ) Verifying Identities Using the Sum or Difference Formulas: Verify the following identities. cos (α − β ) π cos θ − sin θ 10) tan − θ = 9) = 1 + tan α tan β cos α cos β 4 cos θ + sin θ Notes 17b Name ________________________ Double-Angle and Half-Angle Formulas (5.3) Double-Angle Formulas 1. sin 2θ = 2sin θ cos θ 2. cos = 2θ cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 2 tan θ 3. tan 2θ = 1 − tan 2 θ = 1 − 2sin 2 θ Using the Double-Angle Formulas (Forward): Use the given information to find the exact value. 12 1) Find sin 2θ if sin θ = , θ lies in quadrant II. 13 2) Find tan 2θ if cot θ = 3, θ lies in quadrant III. Using the Double-Anlge Formulas (Backwards): Find the exact value of each expression. 2 tan 22.5 π 3) 1 − 2sin 2 4) 1 − tan 2 22.5 12 Verifying Identities Using the Sum or Difference Formulas: Verify the following identities. cos 2 x 2 cot θ 6) sin 2θ = 5) = 1 − tan 2 x 2 1 + cot 2 θ cos x Half-Angle Formulas x 1 − cos x = ± 2 2 1 + cos x x 2. cos = ± 2 2 x 1 − cos x 3. tan = ± 2 1 + cos x 1. sin = 1 − cos x sin x = sin x 1 + cos x Using the Half-Angle Formulas (Forward): Use the given information to find the exact value. 8 θ 7) Find cos if tan θ = − , θ lies in quadrant II. 2 15 8) Find tan θ 2 if sec θ = −3, θ lies in quadrant III. Find the Value: Use a half-angle formula to find the exact value of each expression. 3π 9) sin 22.5 10) tan 8