Calculus Question

Question

Construct two angles in standard position that have sin θ = 24/25, but with different terminal sides. Then find the ratios of the other five trigonometric functions for each angle. Construct the two angles per directions, (there are five additional trig ratios for each angle), and there are six trig ratios considered. Show your work.

Answer 1

24/25 for the sine means a 7-24-25 right triangle. 7squared plus 24 squared = 25 squared

sine = 24/25 in Quadrants I and II. Those two angles are solutions to sin^-1 (24/25) = 73.74 and 106.26 degrees. Just 16.26 degrees either side of 90 degrees A=73.74 B=106.26 Sine is opposite side over hypotenuse. 24/25 is a fairly steep angle, nearer 90 degrees.

A=73.74, B=106.26 sinA=24/25, sinB=24/25 do the inverse sine on a calculator, plug in 24/25 or .96 You should get 73.74 approximately rounded to 2 decimals. Take 90-73.74=16.26, then add 16.26 to 90 to get B=106.26

cosA = 7/25 in Quadrant I but cosB= -7/25 in Quadrant II cosines for A and B are opposite signs

tanA=24/7 in Quadrant I, but tanB=-24/7 in Quadrant II tangents for A and B are opposite signs

secA=25/7, secB=-25/7 secants A and B are opposite sign

cscA=25/24 = cscB also = 25/24 cosecants A and B are the same signs

cotA=7/24 and cotB=-24/7 cotangents for A and B are opposite signs

sine and cosecant are inverses sinA=1/cscA

cosine and secant are inverses cosA=1/secA

tangent and cotangent are inverses tanA=1/cotA

tangent = sine/cosine cotangent = cosine/sine tanA=sinA/cosA cotA=cosA/sinA