Question

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

Answer 1

Given that

`\cos\theta=(√(3))/(2)`

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

`\sin^2\theta+\cos^2\theta=1`

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

`\begin{gathered} \sin^2\theta+((√(3))/(2))^2=1 \\ \sin^2\theta+(3)/(4)=1 \\ \sin^2\theta=(1)/(4) \\ \sin\theta=\pm(1)/(2) \end{gathered}`

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

`\begin{gathered} \tan\theta=(\sin\theta)/(\cos\theta) \\ \sec\theta=(1)/(\cos\theta) \\ \csc\theta=(1)/(\sin\theta) \end{gathered}`

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

`\begin{gathered} \tan\theta=(\pm(1)/(2))/((√(3))/(2))=\pm(1)/(√(3))=\pm(√(3))/(3)\\e-√(3) \\ \\ \csc\theta=(1)/(\pm(1)/(2))=\pm2\\e(1)/(2) \\ \\ \sec\theta=(1)/((√(3))/(2))=(2)/(√(3))=(2√(3))/(3)\\e-2 \\ \\ \sin\theta=\pm(1)/(2)\\e(√(2))/(2) \end{gathered}`

All options are false.