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

User Mskel
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3 votes

Answer:


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)

User Beauchette
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