Question

Given that θ is an obtuse angle and the value of the trigonometric ratio(a) Draw a diagram to model the situation(b) Find cot θ(c) Determine θ to the nearest degree:Cos θ = − 2/9

Answer 1

Explanation:

Given that θ is an obtuse angle and the value of the trigonometric ratio

The equation is given below as

$\cos\theta=-\frac{2}{9}$

Since the angle is an obtuse angle, it greater than 90 but less than 180 degrees which means that the angle falls in the second quadrant

In the second quarant,

$\cos\theta=-ve$

Hence,

The diagram will be given below as

Part b:

To figure out the value of

$\begin{gathered} cot\theta \\ recall: \\ cot\theta=\frac{1}{tan\theta} \\ tan\theta=\frac{sin\theta}{cos\theta} \\ cot\theta=\frac{cos\theta}{sin\theta} \end{gathered}$

To figure value of sin theta we will calulate the opposite using pythagoras theorem below

$\begin{gathered} hyp^2=opp^2+adj^2 \\ 9^2=opp^2+(-2)^2 \\ 81=opp^2+4 \\ opp^2=81-4 \\ opp^2=77 \\ opp=\sqrt{77} \end{gathered}$

Hence,

The value of sin theta will be

$\begin{gathered} sin\theta=\frac{opp}{hyp} \\ \sin\theta=\frac{\sqrt{77}}{9} \end{gathered}$

Hence,

The value of cot theta will be

$\begin{gathered} cot\theta=\frac{cos\theta}{sin\theta} \\ cot\theta=-\frac{2}{9}\times\frac{9}{\sqrt{77}} \\ cot\theta=-\frac{2}{\sqrt{77}}\times\frac{\sqrt{77}}{\sqrt{77}} \\ cot\theta=-\frac{2\sqrt{77}}{77} \end{gathered}$

Hence,

The value of cot theta is

$cot\theta=-\frac{2\sqrt{77}}{77}$

Part C:

To determine the value of theta

$\begin{gathered} cos\theta=-\frac{2}{9} \\ cos\theta=-0.2222 \\ \theta=\cos^{-1}(-0.2222) \\ \theta=102.84^0 \\ to\text{ the nearest degree, we will have} \\ \theta=103^0 \end{gathered}$

Hence,

The value of θ to the nearest degree is

$\theta=103^0$