Question

Suppose tanθ=0.6966.

(a) In what quadrants can the terminal side of θ fall?

(b) Find the possible approximate values of sinθ. Show your work.

(c) For each approximate value of sinθ in part b, find the corresponding approximate value of cosθ. Show your work.

Answer 1

See Below.

Explanation:

A)

Because tanθ > 0, either both sinθ and cosθ are positive or both sinθ and cosθ are negative.

This can only occur in QI or QIII.

B)

From the Pythagorean Identity:

`\displaystyle 1 + \cot^2\theta = \csc^2 \theta`

Solve for sinθ:

`\displaystyle \begin{aligned} 1 + (1)/(\tan^2 \theta) & = (1)/(\sin^2\theta) \\ \\ (1)/(\sin\theta) &= \pm\sqrt{1 + (1)/(\tan^2\theta)} \\ \\ \sin\theta &=\pm\frac{1}{\sqrt{1+(1)/(\tan^2\theta)}} \\ \\ & = \frac{1}{\sqrt{1+(1)/((0.6966)^2)}} \\ \\ & = 0.5716\text{ or } -0.5716 \end{aligned}`

C)

Likewise:

`\displaystyle \begin{aligned} \sin^2\theta+\cos^2\theta & = 1 \\ \\ \cos\theta & = \pm√(1-\sin^2\theta) \\ \\ & = \pm√(1-(\pm0.5716)^2) \\ \\ &= 0.8205\text{ or } -0.8205\end{aligned}`