The coordinates of a point P on the terminal arm of an ∠𝜃 in standard position are given, where 0° ≤ 𝜃 ≤ 360°. Determine the exact values of 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃.

Question

a) 𝑃(4,5)

b) 𝑃(−3,−6)

Answer 1

(4,5)

hypotenuse = square root of (4^2 + 5^2) = sqr(16+25) = sqr41

sin(x) =opposite side over hypotenuse = 5/sqr41 = exactly 5sqr41/41= approximately 0.781

cos(x) =adjacent side over hypotenuse = 4/sqr41 = exactly 4sqr41/41 = approximately 0.625

tan(x) =opposite side over adjacent side 5/4= exactly 1.25

x = about 51.3 degrees

sin51.3 =about 0.781

cos51.3 = about 0.625

tan51.3= 1.25

(-3,-6)

hypotenuse = square root of (9+36) = sqr(45) = 3sqr5

sin(x) =opposite side over hypotenuse = -6/3sqr5 =-2/sqr5 =exactly -2sqr5/5 = about -0.894

cos(x) = adjacent side over hypotenuse = -3/3sqr5 = -1/sqr5 = exactly -sqr5/5 = about -0.447

tan(x) = -6/-3 = exactly 2

x= 63.435 + 180 = 243.435 degrees

sin243.435 = -0.894

cos243.435= -0.447

tan243.435= 2