What is the sine for an angle that has a cosine of 2/5 and is in Quadrant IV? Use the Pythagorean identity sin^2(θ)+cos^2(θ)=1 and the quadrant to solve.(1 point) Responses 21√5 Start Fraction Start Root 21 End Root over 5 End Fraction −21√5 negative Start Fraction Start Root 21 End Root over 5 End Fraction 2125 Start Fraction 21 over 25 End Fraction −2125
To solve this problem, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1.
Since the given angle is in Quadrant IV, the cosine is positive (2/5) and the sine is negative.
To find the sine, we can rearrange the Pythagorean identity to solve for sin(θ):
sin^2(θ) = 1 - cos^2(θ)
sin(θ) = √(1 - cos^2(θ))
Plugging in the given value for cos(θ) = 2/5:
sin(θ) = √(1 - (2/5)^2)
sin(θ) = √(1 - 4/25)
sin(θ) = √(25/25 - 4/25)
sin(θ) = √(21/25)
sin(θ) = √21/5
Since we know the angle is in Quadrant IV and sine is negative in that quadrant, the final answer is:
sin(θ) = -√21/5