Trigonometry Triangles

Question

Hello I currently need help with the following trigonometry question. I believe I need to draw a diagram of a traingle from the information given and then begin solving but I am not sure.

If tan t =−3/4 and π/2 < t< π, find sin t, cos t, sec t, csc t, cot t.

Answer 1

You have the idea. Since t is between 90 and 180, the angle is in quadrant II, where all x coordinates are negative and all y coordinates are positive. The Pythagorean theorem tells you that the hypotenuse will be 5 with the horizontal leg as -4 and vertical leg as 3, i.e. the tangent is -3/4. Also you might recognize this as a 3-4-5 right triangle without having to use the Pythagorean Theorem. Now you can deduce the other 5 trig ratios. For example, sin t = 3/5 (opposite/hypotenuse).

Answer 2

Your instinct to draw a triangle is exactly right, so start there. The other thing to pay attention to is that interval, pi/2 < t < pi, because that is telling you the angle sits in quadrant II, and that is what decides all the signs at the end.

We know tan t = -3/4, and tangent is opposite over adjacent. Since we are in quadrant II, x is negative and y is positive there, so picture a right triangle drawn against the negative x-axis with an opposite side of 3 and an adjacent side of -4. That gives us tangent of 3/(-4), which is the -3/4 we were given.

Now we need the hypotenuse, and we get it from the Pythagorean theorem. The hypotenuse is always positive, since it is a distance:

r^2 = 3^2 + (-4)^2 = 9 + 16 = 25

r = 5

If you notice, this is just the 3-4-5 triangle, which shows up constantly in these problems, so it is worth recognizing on sight.

Now we can read off everything, using sine as opposite over hypotenuse and cosine as adjacent over hypotenuse:

sin t = 3/5

cos t = -4/5

The other three are just the reciprocals of these, so we flip each fraction:

sec t = 1/cos t = -5/4

csc t = 1/sin t = 5/3

cot t = 1/tan t = -4/3

Remember, our angle is in quadrant II, so let us sanity check the signs. In quadrant II sine is positive and cosine is negative, which matches what we got, and since secant and cotangent come from cosine and tangent they both stay negative while cosecant stays positive.

So, our final answers are sin t = 3/5, cos t = -4/5, sec t = -5/4, csc t = 5/3, and cot t = -4/3.