Identidades Trigonométricas, Esquemas de Cálculo

Baixe Identidades Trigonométricas e outras Esquemas em PDF para Cálculo, somente na Docsity! Trigonometric Identities Pythagoras’s theorem sin2 θ + cos2 θ = 1 (1) 1 + cot2 θ = cosec2θ (2) tan2 θ + 1 = sec2 θ (3) Note that (2) = (1)/ sin2 θ and (3) = (1)/ cos2 θ. Compound-angle formulae cos(A+B) = cosA cosB − sinA sinB (4) cos(A−B) = cosA cosB + sinA sinB (5) sin(A+B) = sinA cosB + cosA sinB (6) sin(A−B) = sinA cosB − cosA sinB (7) tan(A+B) = tanA+ tanB 1− tanA tanB (8) tan(A−B) = tanA− tanB 1 + tanA tanB (9) cos 2θ = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1− 2 sin2 θ (10) sin 2θ = 2 sin θ cos θ (11) tan 2θ = 2 tan θ 1− tan2 θ (12) Note that you can get (5) from (4) by replacing B with −B, and using the fact that cos(−B) = cosB (cos is even) and sin(−B) = − sinB (sin is odd). Similarly (7) comes from (6). (8) is obtained by dividing (6) by (4) and dividing top and bottom by cosA cosB, while (9) is obtained by dividing (7) by (5) and dividing top and bottom by cosA cosB. (10), (11), and (12) are special cases of (4), (6), and (8) obtained by putting A = B = θ. Sum and product formulae cosA+ cosB = 2 cos A+B 2 cos A−B 2 (13) cosA− cosB = −2 sin A+B 2 sin A−B 2 (14) sinA+ sinB = 2 sin A+B 2 cos A−B 2 (15) sinA− sinB = 2 cos A+B 2 sin A−B 2 (16) Note that (13) and (14) come from (4) and (5) (to get (13), use (4) to expand cosA = cos(A+B2 + A−B 2 ) and (5) to expand cosB = cos( A+B 2 − A−B 2 ), and add the results). Similarly (15) and (16) come from (6) and (7). Thus you only need to remember (1), (4), and (6): the other identities can be derived from these.