If tan t = −7/24 and 3 π/2 < t < 2 π , find sin t , cos t , sec t , csc t , cot t . Enter the exact answers.

Question

sin t=

cos t=

sec t=

csc t=

cot t=

Answer 1

since t falls between 3 π/2 and 2 π, t is in the 4th quadrant. Draw a right triangle with the hypotenuse starting at the origin and forming an angle of t degrees with the x axis. since we know tan (t) is -7/24, that means opposite of t is -7 and adjacent of t is 24. We know the 7 must be negative instead of the 24 because it is in the 4th quadrant so the y value will be negative, and x values will be positive. Now use pythagorean theorem to find the missing side, the hypotenuse. 7,24,25 is a pythagoean triple, so we know the missing side is 25. If you did not recognize the pythagorean triple, you can use a^2+b^2=c^2 where a is -7, b is 24, and solve for c. (-7)^2+(24)^2=c^2. c=25. now we know that opposite is -7, adjacent is 24, and hypotenuse is 25.

sin t=opp/hyp=-7/25

cos t=adj/hyp=24/25

sec t=hyp/adj=25/24

csc t=hyp/opp=-25/7

cot t=adj/opp=-24/7

I hope this helps!

Answer 2

Make a right triangle in the 4th quadrant.

Use the Pythagorean theorem to find the hypotenuse.

Then it's just a matter of writing each of the other trig ratios...