Solve 4 sin ² ( t ) - 3 sin ( t /3)= 1 .

Solution

Before solving this equation let's solve clearly unrelated equation.

4x² - 3x = 1  ⇒ 4x² - 3x -1 = ( 4x + 1) ( x -1) = 0 ⇒   x= - 1 ,1

It is an easy equation to solve out. Then the obvious question is, why did we do this?  We'll, if you compare the two equations you'll illustrates that the only real difference is that the one we just solved has an x everywhere & the equation we desire to solve has a sine.  What this tells us is that we can work the two equations in accurately the same way.

We, will first "factor" the equation as follows,

4 sin ² ( t /3) - 3sin ( t/3 ) -1 = ( 4 sin ( t /3) + 1) ( sin ( t /3) -1) = 0

Now, set each of the two factors equivalent to zero and solve for the sine,

sin ( t /3) = - 1/4                               sin ( t/3 )= 1

Now we have two trig equations which we can easily (hopefully...) solve at this point.  We'll leave the details to you to check out that the solutions to each of these & so the solutions to the original equation are following,

t= 18.0915 + 6 ∏ n

t= 10.1829 + 6 ∏ n     n= 0, ±1, ±2,..........

t= 3 ∏/2 + 6 ∏ n

The first two solutions are from the first equation & the third solution is from the second equation.

Let's see more trig equation which involves solving a quadratic equation.  Although, this time, unlike the earlier example this one won't factor and thus we'll have to use the quadratic formula.