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Question

Write the letter of the correct answer. If

sinθ=23\sin \theta = \frac { 2 } { 3 }

, then

cos2θ=\cos 2 \theta =

__.

Solution

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Answered 2 years ago
Answered 2 years ago
Step 1
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Given in problem : sinθ=23\sin\theta=\dfrac{2}{3}

To find the value of cos2θ\cos 2\theta we will use formula cos2θ=cos2θsin2θ\cos2\theta=\cos^2\theta-\sin^2\theta so we need first find the value of cos2theta\cos^2 theta. So we have :

cos2θ=1sin2θPythagorean identity sin2x+cos2x=1=1(23)2Putting the value=949Taking LCM=59Simplifying\begin{align*} \cos^2 \theta&=1-\sin^2\theta &&\text{\color{#4257b2}Pythagorean identity $\sin^2x+\cos^2x=1$}\\ \\ &=1-\left(\dfrac{2}{3}\right)^2&&\text{\color{#4257b2}Putting the value}\\ \\ &=\dfrac{9-4}{9} &&\text{\color{#4257b2}Taking LCM}\\ \\ &=\dfrac{5}{9} &&\text{\color{#4257b2}Simplifying} \end{align*}

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