Question

50 points need help asap plzz!!! Show that the Pythagorean identity sin^2 θ+cos^2 θ=1 is true for the given angle.
θ= 5π/3

Answer 1

Answer:

Pythagorean identity sin²θ+cos²θ = 1 is true for the angle θ = $\frac{5\pi }{3}$

Step-by-step explanation:

At first, let us simplify the left side of the identity

∵ The left side is sin²Ф + cos²Ф

∵ Ф = $\frac{5\pi }{3}$ ⇒ lies in the 4th quadrant

∴ The left side is sin²($\frac{5\pi }{3}$) + cos²($\frac{5\pi }{3}$)

→ Let us write the values of sin($\frac{5\pi }{3}$) and cos($\frac{5\pi }{3}$)

∵ sin($\frac{5\pi }{3}$) = $\frac{-\sqrt{3}}{2}$ ⇒ sine an angle in the 4th quadrant is -ve

∵ cos($\frac{5\pi }{3}$) = $\frac{1}{2}$ ⇒ cosine an angle in the 4th quadrant is +ve

→ Substitute them in the left side

∵ sin²($\frac{5\pi }{3}$) + cos²($\frac{5\pi }{3}$) = [ $\frac{-\sqrt{3}}{2}$]² + [$\frac{1}{2}$]²

∴ sin²($\frac{5\pi }{3}$) + cos²($\frac{5\pi }{3}$) = [$\frac{3}{4}$] + [$\frac{1}{4}$]

∴ sin²($\frac{5\pi }{3}$) + cos²($\frac{5\pi }{3}$) = [$\frac{4}{4}$]

∴ sin²($\frac{5\pi }{3}$) + cos²($\frac{5\pi }{3}$) = 1

∵ The right side = 1

∴ Left side = Right side

∴ sin²($\frac{5\pi }{3}$) + cos²($\frac{5\pi }{3}$) = 1 ⇒ proved

∴ Pythagorean identity sin²θ+cos²θ = 1 is true for the angle θ = $\frac{5\pi }{3}$