Question

4. Solve the following for 0° 50 360°. (3 marks each) sin + - = 0 a) 3 4 b) cos 0 = 0.276

Answer 1

Given

a ) sin( θ ) + 3 / 4 = 0

0 < = θ < = 360

=> sin( θ ) = - ( 3 / 4 )

=> θ = arcsin( - 3 / 4 )

We know that --------------> arcsin( - A ) = - arcsin( A ) ............... Trigonometric Identity

=> arcsin( - 3 / 4 ) = - arcsin( 3 / 4 ) + 360*( n )

We know that ------> 0 < = θ < = 360

Put n = 1

θ = - arcsin( 3 / 4 ) + 360*( 1 )

=> θ = - arcsin( 3 / 4 ) + 360*( 1 )

=> θ = - 48.59 + 360

=> θ = 311.41 degree

We know that --------------> arcsin( - A ) = 180 + arcsin( A ) ............... Trigonometric Identity

=> arcsin( - 3 / 4 ) = 180 + arcsin( 3 / 4 ) + 360*( n )

We know that ------> 0 < = θ < = 360

Put n = 0

θ = 180 + arcsin( 3 / 4 ) + 360*( 0 )

=> θ = 180 + arcsin( 3 / 4 )

=> θ = 180 + 48.59

=> θ = 228.59 degree

So,  θ = 228.59 degree , 311.41 degree

b ) cos( θ ) = 0.276

0 < = θ < = 360

=> θ = arccos( 0.276 ) + 360*( n )

=> θ = 73.97 + 360*( n )

We know that ------> 0 < = θ < = 360

Put n = 0

=> θ = 73.97 + 360*( 0 )

=> θ = 73.97 degree

We know that -------------> cos( θ ) = cos( 360 - θ ) ...................... Trigonometric Identity

=> cos( 360 - θ ) = 0.276

=> 360 - θ = arccos( 0.276 )

=> θ = 360 - arccos( 0.276 ) + 360*( n )

=> θ = 360 - 73.97 + 360*( n )

=> θ = 286.03 + 360*( n )

Put n = 0

=> θ = 286.03 + 360*( 0 )

=> θ = 286.03 degree

So,  θ = 73.97 degree , 286.03 degree