Nội dung text 38 Trigonometric Identites.pdf
tan 2θ = tan(θ + θ) = tan θ + tan θ 1 − tan θ tan θ tan 2θ = 2 tan θ 1 − tan2 θ Furthermore, for cos 2θ, cos 2θ = cos2 θ − sin2 θ = (1 − sin2 θ) − sin2 θ cos 2θ = 1 − 2 sin2 θ cos 2θ = cos2 θ − sin2 θ = cos2 θ − (1 − cos2 θ) cos 2θ = 2 cos2 θ − 1 Double Angle Identities. sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1 tan 2θ = 2 tan θ 1 − tan2 θ 5. Complementary Identities Some complementary identities can be derived from the sum identities. Note that the angle complementary to θ is 90° − θ, sin(90° − θ) = sin ⏞ 90° 1 cos θ − cos ⏞ 90° 0 sin θ sin(90° − θ) = cos θ Similarly, cos(90° − θ) = cos ⏞ 90° 0 cos θ + sin ⏞ 90° 1 sin θ cos(90° − θ) = sin θ For the other functions, tan θ = sin θ cos θ tan(90° − θ) = sin(90° − θ) cos(90° − θ) tan(90° − θ) = cos θ sin θ tan(90° − θ) = cot θ
csc θ = 1 sin θ csc(90° − θ) = 1 sin(90° − θ) csc(90° − θ) = 1 cos θ csc(90° − θ) = sec θ Summarizing, Complementary Identities. sin(90° − θ) = cos θ cos(90° − θ) = sin θ csc(90° − θ) = sec θ sec(90° − θ) = csc θ tan(90° − θ) = cot θ cot(90° − θ) = tan θ 6. Product Identities From the sum identity, cos(α + β) = cos α cos β − sin α sin β cos(α − β) = cos α cos β + sin α sin β Adding the two equations, cos(α + β) + cos(α − β) = 2 cos α cos β cos αcos β = 1 2 [cos(α + β) + cos(α − β)] Subtracting the two equations, cos(α + β) − cos(α − β) = −2 sin α sin β sin αsin β = 1 2 [cos(α − β) − cos(α + β)] From the sum identity, sin(α + β) = sin α cos β + cos α sin β sin(α − β) = sin α cos β − cos α sin β Adding the two equations, sin(α + β) + sin(α − β) = 2 sin α cos β sin αcos β = 1 2 [sin(α + β) + sin(α − β)] Subtracting the two equations, sin(α + β) − sin(α − β) = 2 cos α sin β cos αsin β = 1 2 [sin(α + β) − sin(α − β)]