Question

Find the exact values of sin2 θ for cos θ = 3/18 on the interval 0° ≤ θ ≤ 90°

Answer 1

`sin(2\theta)=(√(35) )/(18)`

Explanation:

Recall the formula for the sine of the double angle:

`sin(2\theta)=2*sin(\theta)*cos(\theta)`

we know that
`cos(\theta)=(3)/(18)`, and that
`\theta` is in the interval between 0 and 90 degrees, where both the functions sine and cosine are non-negative numbers. Based on such, we can find using the Pythagorean trigonometric property that relates sine and cosine of the same angle, what
`sin(\theta)` is:

`cos^2(\theta)+sin^2(\theta)=1\\sin^2(\theta)=1-cos^2(\theta)\\sin(\theta)=√(1-cos^2(\theta)) \\sin(\theta)=\sqrt{1-((3)/(18) )^2}\\sin(\theta)=\sqrt{1-(9)/(324) }\\sin(\theta)=\sqrt{(324-9)/(324) }\\sin(\theta)=\sqrt{(315)/(324) }\\\\sin(\theta)=(3)/(18)√(35 )`

With this information, we can now complete the value of the sine of the double angle requested:

`sin(2\theta)=2*sin(\theta)*cos(\theta)\\sin(2\theta)=2*(3)/(18) \,√(35) \,(3)/(18)\\sin(2\theta)=(2*3*3)/(18*18)\,√(35) \\sin(2\theta)=(√(35) )/(18)`