Solve the trigonometric equation6sin(𝑡2)−2=−5to find an exact solution on the interval [−𝜋2,𝜋2]

To solve the trigonometric equation 6sin(t/2) - 2 = -5, follow these steps:

Step 1: Isolate the sine function. To do this, add 2 to both sides of the equation to get: 6sin(t/2) = -5 + 2 6sin(t/2) = -3

Step 2: Divide both sides by 6 to solve for sin(t/2): sin(t/2) = -3/6 sin(t/2) = -1/2

Step 3: Find the angle whose sine is -1/2. This is where the unit circle comes in handy. The sine of an angle is negative in the third and fourth quadrants. However, since we are looking for solutions in the interval [-π/2, π/2], we are only interested in the fourth quadrant. The angle in the fourth quadrant whose sine is 1/2 is -π/6.

So, t/2 = -π/6

Step 4: Multiply both sides by 2 to solve for t: t = 2*(-π/6) t = -π/3

So, the solution to the equation 6sin(t/2) - 2 = -5 in the interval [-π/2, π/2] is t = -π/3.