Trig Equations with Non-Standard Angles Examples

Solve 3sin (t + π) = 2 on the interval .

t + π = 0.730 radians

Step one is to isolate the trig function. Step two is to use the inverse function to get rid of the sine surrounding t. Step three is to stop: Hammer-time.

0.730 radians lies in the first quadrant. Another angle, that will share the same value for sine, will be located in the second quadrant (where sine is also positive). Its corresponding acute angle will be 0.730 radians; that angle is (π – 0.730) = 2.412 radians.

t + π = 0.730 + 2πn
t + π = 2.412 + 2πn

t = -2.412 + 2πn
t = -0.730 + 2πn

Now we find all of the coterminal angles that satisfy the equality.

If we add or subtract 2π (n = \pm 1), then the angles will be positive or less than 2π, respectively. So, our solutions in the interval are:

t = -2.412 and -0.730 radians

Solve 0.9cos t + (cos t)(sin t) = 0.

Sometimes, a little bit of work needs to be done before we can start solving trig equations.

(cos t)(0.9 + sin t) = 0

In this case, we need to factor out the common term, cos t.

If either factor equals zero, the whole equation will.

cos t = 0
0.9 + sin t = 0

Cosine equals zero for 0 and π radians on the unit circle, and all of their coterminal angles.

t = 0 + 2πn
t = π + 2πn

OR

t = πn

Now we can find angles for sine.

sin t = -0.9

sin^-1 (sin t) = sin^-1 -0.9

t = -1.120 radians

That's only one angle though. -1.12 radians is in the fourth quadrant. Sine of an angle in the fourth quadrant is negative, so our other angle must be in the third quadrant.

π + 1.120 = 4.261 radians

t = -1.120 + 2πn
t = 4.261 + 2πn

These are the infinite number of angles that will make sine equal -0.9. However, we really wanted the angles that satisfied our original equation (waaaaay up top there). Including those from cosine, they are:

t = πn
t = -1.120 + 2πn
t = 4.261 + 2πn

Finding the right answer satisfies us almost as much as it does the equation.

Solve csc t = 3sec t.

Remember what cosecant and secant actually mean.

Because of their flip-flopped nature, if we multiply through by cosine we get:

cot t = 3

Which itself equals:

t = 0.322 radians

This is our first angle, and our corresponding acute angle for the other angle.

t = 0.322 + 2πn
t = 3.463 + 2πn