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If cosθ=3√2cosθ=32 then which of the following could be true?tan=−3√tangent is equal to negative square root of 3cscθ=12cosecant theta is equal to 1 halfsecθ=−2secant theta is equal to negative 2sinθ=2√2sine theta is equal to the fraction with numerator square root of 2 and denominator 2

If Cosθ32cosθ32 Then Which Of The Following Could Be Truetan3tangent Is Equal To Negative Square Root Of 3cscθ12cosecant Theta Is Equal To 1 Halfsecθ2secant The class=

Sagot :

Given that

[tex]\cos\theta=\frac{\sqrt{3}}{2}[/tex]

we can determinate the sine of this angle using the following identity

[tex]\sin^2\theta+\cos^2\theta=1[/tex]

If we substitute the value of the cosine on this identity, we're going to have:

[tex]\begin{gathered} \sin^2\theta+(\frac{\sqrt{3}}{2})^2=1 \\ \sin^2\theta+\frac{3}{4}=1 \\ \sin^2\theta=\frac{1}{4} \\ \sin\theta=\pm\frac{1}{2} \end{gathered}[/tex]

The definitions of secant, tangent, and cosecant in terms of the sine and cosine are given by:

[tex]\begin{gathered} \tan\theta=\frac{\sin\theta}{\cos\theta} \\ \sec\theta=\frac{1}{\cos\theta} \\ \csc\theta=\frac{1}{\sin\theta} \end{gathered}[/tex]

Using the known values for the sine and cosine functions on those definitions, we have:

[tex]\begin{gathered} \tan\theta=\frac{\pm\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\pm\frac{1}{\sqrt{3}}=\pm\frac{\sqrt{3}}{3}\ne-\sqrt{3} \\ \\ \csc\theta=\frac{1}{\pm\frac{1}{2}}=\pm2\ne\frac{1}{2} \\ \\ \sec\theta=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\ne-2 \\ \\ \sin\theta=\pm\frac{1}{2}\ne\frac{\sqrt{2}}{2} \end{gathered}[/tex]

All options are false.

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