Question

Using the identity sin^ 2 © + cos ^2 © = 1, find the value of tan O, to the nearesthundredth, if cos 0 = -0.27 and pie < 0 < 3pie/ 2

Answer 1

Given the identity:

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

Let's find the value of tanθ, where:

`cos\theta=-0.27\text{ and }\pi<\theta<(3\pi)/(2)`

Since it is in the given interval, θ is in the third quadrant.

Now, substitute -0.27 for θ in the identity:

`\begin{gathered} sin^2\theta+(-0.27)^2=1 \\ \\ sin^2\theta+0.0729=1 \\ \\ \end{gathered}`

Subtract 0.0729 from both sides:

`\begin{gathered} sin^2\theta+0.0729-0.0729=1-0.0729 \\ \\ sin^2\theta=0.9271 \end{gathered}`

Take the square root of both sides:

`\begin{gathered} sin\theta=√(0.9271) \\ \\ sin\theta=0.9629 \end{gathered}`

Now, apply the trigonometric identity:

`\begin{gathered} tan\theta=(sin\theta)/(cos\theta) \\ \\ tan\theta=(0.9629)/(-0.27) \\ \\ tan\theta=−3.57 \end{gathered}`

Since tan is positive in the third quadrant, we have:

`tan\theta=3.57`

ANSWER:

3.57