If sinθ=1/5 and θis in Quadrant I, find the exact value of 2θ.
Use the double angle formula:
sin(2θ)=2sin(θ)cos(θ)
given
sin(θ)=1/5
cos(θ)=sqrt(1-(1/5)²))=sqrt(24)/5
To find the exact value of 2θ, we first need to determine the value of θ.
Given that sinθ = 1/5 and θ is in Quadrant I, we can conclude that θ is an acute angle in the right-angled triangle. The opposite side of the triangle is 1, and the hypotenuse is 5.
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the adjacent side of the triangle:
a^2 + 1^2 = 5^2
a^2 + 1 = 25
a^2 = 24
a = √24
a = 2√6
Now that we know the values of the different sides of the triangle, we can find the value of θ. From the definition of the sine function (sinθ = opposite/hypotenuse), we can write:
sinθ = 1/5
opposite/hypotenuse = 1/5
1/5 = 1/5
Since we have already determined that the opposite side is 1 and the hypotenuse is 5, the value of θ can be found as the inverse sine (or arcsine) of 1/5:
θ = arcsin(1/5)
θ ≈ 11.54° (rounded to two decimal places)
Now we can find the value of 2θ:
2θ = 2 * 11.54°
2θ ≈ 23.08°
Therefore, the exact value of 2θ is approximately 23.08°.