Solve for t in (0,2pi)

**sin 2t - 3 = 6sin t - cost**

### 2 Answers | Add Yours

We'll move all terms of the equation, to the left side:

sin 2t - 3 - 6 sin t + cost = 0

We'll re-write the equation, substituteing sin2t = 2sintcost

2sintcost - 6 sin t + cost - 3 = 0

We could also factorize the first term and the3rd term, by cos t:

cos t(2sin t+1) - 6 sin t - 3 = 0

We'll also factorize the last 2 terms by -3:

cos t(2sin t+1) - 3(2sin t + 1) = 0

We'll factorize again by 2sin t+1:

(2sin t+1)(cos t - 3) = 0

We'll set the first factor as zero:

2sin t + 1 = 0

We'll subtract 1:

2sin t = -1

sin t = -1/2

t = arcsin (-1/2)

Since we have to solve the equation in the range (0,2pi), we'll validate the solution from the unit circle.

The sine function is negative in the 3rd and 4th quadrant, so the solutions for t are:

t = pi + pi/6

t = 7pi/6 (3rd quadrant)

t = 2pi - pi/6

t = 11pi/6 (4th quadrant)

We'll set the 2nd factor as zero:

cos t - 3 = 0

We'll add 3 both sides:

cost = 3

There are no solutions for t in this case, since the value of the function cosine is not bigger than 1.

**There are only 2 valid solutions for the given equation, in the range (0,2pi): {7pi/6 ; 11pi/6}.**

You need to use the double angle identity, such that:

`sin 2t = 2 sin t cos t`

You need to write the equation in terms of `sin t` and `cos t` , such that:

`2 sin t cos t - 3 = 6 sin t - cos t`

You need to group the terms such that:

`2 sin t cos t - 6 sin t = 3 - cos t`

Factoring out 2sin (t) to the left side yields:

`2sin (t) (cos (t) - 3 ) = -(cos (t) - 3) => 2sin (t) (cos (t) - 3) + (cos (t) - 3) = 0`

You need to factor out (`cos t - 3)` such that:

`(cos t - 3)(2sin t + 1) = 0 => {(cos t - 3 = 0),(2 sin t + 1 = 0):}`

`cos t = 3` invalid , `cos t in [-1,1]`

`2 sin (t) = -1 => sin (t) = -1/2`

Sine is negative for `t in (pi,(3pi)/2) U ((3pi)/2, 2pi)` , such that:

`t = pi + pi/6 => t = (7pi)/6`

`t = 2pi - pi/6 => t = (11pi)/6`

**Hence, evaluating the solutions to the given equation, under the given conditions, yields **`t = (7pi)/6 ; t = (11pi)/6.`

### Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes