Assume cos(t) = (-17)/(95) where (π)/(2) < t

Assume $cos(t) = \frac{-17}{95}$ where $\frac{\pi}{2} < t < \pi$. Compute the following: $cos(-t) = \frac{-17}{95}$ $sin(t) = \frac{4\sqrt{546}}{95}$ $sin(-t) = \frac{-4\sqrt{546}}{95}$ $sin(-t) + cos(-t) = \frac{-17 + 4\sqrt{546}}{95}$ $sin^2(-t) + cos^2(-t) = 1$ $sin(t - \frac{\pi}{2}) = $cos(t + \frac{\pi}{2}) = \frac{34 - 95\pi}{190}$

Transcript

00:01 This problem, it is said that tan t is equal to negative 3 over 4, and pi over 2 is less than t is less than pi, and we need to find the other trigonometric functions.

00:14 So, first of all, let us determine sec t.

00:19 Now, 1 plus tan square t, this will give us sec square t, using the pythagorean identity.

00:28 And we want to find sec t, so we're going to take the square root.

00:32 Take the positive or negative square root.

00:34 Well, pi over 2 is less than t is less than pi, meaning that t is in the second quadrant.

00:40 And in the second quadrant, sec is negative.

00:42 So we have negative the square root of 1 plus 10 square root of 1 plus negative 3 over 4 whole squared.

00:50 So negative square root of 1 plus 9 over 16.

00:56 That's negative the square root of 16 plus 9, which is 25 divided by 16.

01:01 So we have.

01:02 Negative 5 over 4 and so that is the value of sec t now we can determine cost t as one divided by sec t so that will be one divided by negative 5 over 4 so this is negative 4 over 5 so that's cost t now let us determine sine t so what is sign t well sine t can be obtained as tan t times cost t this is because tan t is sine t over cost t and tan t is negative 3 over 4, and we calculated cost t to be negative 4 over 5.

01:37 So we are left with 3 over 5, that is sine t.

01:41 And we can use that to find kosek t...

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