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

Step-by-step explanation:

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

The equation is given below as

`\cos\theta=-(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=(1)/(tan\theta) \\ tan\theta=(sin\theta)/(cos\theta) \\ cot\theta=(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=√(77) \end{gathered}`

Hence,

The value of sin theta will be

`\begin{gathered} sin\theta=(opp)/(hyp) \\ \sin\theta=(√(77))/(9) \end{gathered}`

Hence,

The value of cot theta will be

`\begin{gathered} cot\theta=(cos\theta)/(sin\theta) \\ cot\theta=-(2)/(9)*(9)/(√(77)) \\ cot\theta=-(2)/(√(77))*(√(77))/(√(77)) \\ cot\theta=-(2√(77))/(77) \end{gathered}`

Hence,

The value of cot theta is

`cot\theta=-(2√(77))/(77)`

Part C:

To determine the value of theta

`\begin{gathered} cos\theta=-(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`