Georgina is told that and angle has a sine of √5/5 and is in quadrant II. use the Pythagorean identity sin^2(0) + cos^2(0) = 1, the trigonometric identity tan0 = sin0/cos0 and the quadrant to find the tangent.

Answer 1

To find the tangent of the angle, we need to determine the cosine of the angle and then use the trigonometric identity tan(θ) = sin(θ)/cos(θ).

Given that the sine of the angle is √5/5, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to find the cosine of the angle.

sin^2(θ) + cos^2(θ) = 1
(√5/5)^2 + cos^2(θ) = 1
5/25 + cos^2(θ) = 1
cos^2(θ) = 1 - 5/25
cos^2(θ) = 20/25
cos^2(θ) = 4/5
cos(θ) = ±√(4/5)

Since the angle is in quadrant II, the cosine is negative.

cos(θ) = -√(4/5)

Now, using the trigonometric identity tan(θ) = sin(θ)/cos(θ), we can find the tangent.

tan(θ) = sin(θ)/cos(θ)
tan(θ) = (√5/5)/(-√(4/5))
tan(θ) = (√5/5) * (-√(5/4))
tan(θ) = -√((5/5) * (5/4))
tan(θ) = -√(25/20)
tan(θ) = -√(5/4)
tan(θ) = -√5/√4
tan(θ) = -√5/2

Therefore, the tangent of the angle is -√5/2.