θ is an angle in standard position. P(5, -12) is a point on the terminal arm of θ. Determine the exact value of (cotθ)(secθ)+(sinθ). Include a diagram

Question

URGENT!

Answer 1

Whenever we are given a point on the terminal ray of an angle, we can calculate all 6 trig values of that angle (without ever knowing, btw, the exact measure of the angle) as follows:

Use pythagorean theorem to calculate r: x² + y² = r² so ... 5² + (-12)² = r² and r = 13

Use coordinate plane definitions of the trig ratios:

sinΘ = y/r ; cscΘ = r/y so sinΘ = - 12/13 and cscΘ = - 13/12

cosΘ = x/r ; secΘ = r/x. so cosΘ = 5/13 and secΘ = 13/5

tanΘ = y/x ; cotΘ = x/y so tanΘ = - 12/5 and cotΘ = - 5/12

Finally, cotΘ·secΘ + sinΘ = (-5/12)(13/5) + (-12/13) = -13/12 - 12/13 = - 313/156