Prove that:
(1) sin2/ cos + cos = (sin2+cos2)/cos= 1/cos [sin2+cos2 = 1]= sec [1/cos = sec]Hence proved. (2) cos2 (1 + tan2 ) = cos2 +sin2 [cos2 ×tan2 = cos2 ×sin2/cos2 = sin2]= 1 [sin2+cos2 = 1]Hence proved. (3) √[(1-sin)/(1+ sin)] = √[(1-sin)/(1+ sin)]×√[(1- sin)/(1- sin)] [rationalizing denominator]= √[(1-sin)2/(1-sin2)]= √[(1-sin)2/cos2 [1-sin2 = cos2 ]= (1-sin)/cos [taking square root]= (1/cos)-(sin/cos)= sec-tan [1/cos = sec , sin/cos = tanHence proved. (4) […]
If tan = 1 then, find the values of (sin+ cos)/( sec+ cosecθ ).
Given tan = 1We know that tan 45˚ = 1= 45˚sin 45 = 1/√2cos45 = 1/√2sec45 = √2cosec45 = √2(sin+ cos)/( sec+ cosecθ ) = (sin45+ cos45)/( sec45+ cosec45 )= [(1/√2)+( 1/√2)]÷[√2+√2]= (2/√2)÷2√2= (2/√2)×(1/2√2)= 1/2Hence (sin+ cos)/( sec+ cosecθ ) = 1/2
If 5sec- 12cosec = 0, find the values of sec, cos and sin.
Given 5sec- 12cosec = 05sec = 12cosec5/cos = 12/sin [sec = 1/cos and cosec = 1/sin]5/12 = cos/sinsin/cos = 12/5tan = 12/5We know that 1+tan2 = sec21+(12/5)2 = sec21+(144/25) = sec2(25+144)/25 = sec2169/25 = sec2Taking square root on both sidessec = 13/5cos = 1/sec = 5/13We know that sin2+cos2=1sin2+(5/13)2 = 1sin2 = 1-(5/13)2sin2 = 1-(25/169)sin2 = (169-25)/169sin2 = 144/169Taking square root on […]
If cot = 40/9 , find the values of cosec and sin.
Given cot = 40/9We have 1+cot2 = cosec21+(40/9)2 = cosec21+(1600/81) = cosec2(81+1600)/81 = cosec21681/81 = cosec2cosec2 = 1681/81Taking square root on both sidescosec = 41/9We have sin = 1/cosecsin = 1÷(41/9)sin = 9/41Hence cosec = 41/9 and sin = 9/41.
If tan = 3/4 , find the values of sec and cos.
Given tan = 3/4We have 1+tan2= sec21+(3/4)2 = sec21+(9/16) = sec2sec2 = (16+9)/16 = 25/16Taking square root on both sidessec = 5/4We have cos = 1/seccos = 1÷(5/4)cos = 4/5Hence sec = 5/4 and cos = 4/5.
If sin = 7/25 , find the values of cos and tan.
Given sin = 7/25We have sin2+cos2 = 1(7/25)2+ cos2 = 1(49/625)+ cos2 = 1cos2 = 1-(49/625)cos2 = (625-49)/625cos2 = 576/625Taking square root on both sidescos = 24/25tan = sin/cos= (7/25) ÷(24/25)= (7/25) ×(25/24)= 7/24Hence cos = 24/25 and tan = 7/24.