Solve the trigonometric equation Sin x 0493 for 0 degree l

Solve the trigonometric equation Sin x = - 0.493 for 0 degree lessthanorequalto x lessthanorequalto 360. degree Give your answers correct to one decimal place. Solve the equation 3 Cos^2 theta - 2 Sin theta - 2 = 0 for 0 degree lessthanorequalto x lessthanorequalto 360. degree Give your answers correct to one decimal place. Solve the trigonometric equation 5 Sin theta + 3 = 0. Give your answer correct to 2 decimal places. Solve the hyperbolic equation 3 Sinh x + 2 Cosh x = 0 giving your answer correct to 4 decimal places. Solve the equation 2.6 chx + 5.1 shx = 8.73 correct to 4 decimal places. Prove the identity Sin^2 theta. Cot theta. Sec theta = Sin theta. Prove that 1 + Cot theta/1 + Tan theta = Cot theta. Show that Cos^2 theta - Sin^2 theta = 1 - 2 Sin^2 theta.

Solution

a. Sin x=-.493

Taking sin^-1 on both sides

sin^-1(sin x)=sin^-1-0.493

x=-29.54degree,209.54 degree,330.46 degree

and x have to be between 0 and 360

Therefore x=209.54 degree and 330.46 degree

b. 3 cos^2theta - 2 sin theta-2=0

cos^2 theta=1-sin^2 theta

3(1-sin^2 theta)-2sin theta -2=0

Let sin theta=t

3(1-t^2)-2t-2=0

3-3t^2 -2t-2=0

3t^2+2t-1=0

(3t-1)(t+1)=0

t=1/3,-1

Substituting back sin theta for t

sin theta=1/3 and sin theta=-1

Taking inverse on both sides

sin^-1(sin theta)=sin^-1(1/3) and sin^-1(sin theta)=sin^-1(-1)

theta= 19.5 degree, 160.5 degree and theta=-90degree, 270 degree

theta have to be greater than 0

therefore theta=19.5degree,160.5degree,270 degree

c. 5 sin theta+3=0

sin theta=-3/5

taking inverse on both sides

sin^-1(sin theta)=sin^-1(-3/5)

theta=sin^-1(-3/5)

theta=-36.87 degree,216.87degree, 323.13degree

And for theta lies between 0 and 360

theta=216.87 degree and 323.13 degree