Question

Finding trigonometric ratios from a point on the unit circle

Answer 1

`\begin{gathered} co\sec \theta\text{ = }\frac{2}{-\text{ }\sqrt[]{2}} \\ \\ \sin \theta\text{ = }\frac{-\text{ }\sqrt[]{2}}{2} \\ \\ \cot \theta\text{ = }1 \end{gathered}`Step-by-step explanation:

The terminal side of theta intersects with unit circle at (-√2/2, -√2/2)

The only quadrant where we have (-, -), that is the y and x coordinate is negative is the third quadrant

In the third quadrant: tan is positive, cos is negative and sine is negative

from the point of the unit circle:

`\begin{gathered} x\text{ = adjacent = }\frac{-\sqrt[]{2}}{2} \\ y\text{ = opposite = }\frac{-\sqrt[]{2}}{2} \\ \text{hypotenuse}^2=opposite^2+adjacent^2 \\ \text{hypotenuse}^2=\text{ }(\frac{-\sqrt[]{2}}{2})^2\text{ + }(\frac{-\sqrt[]{2}}{2})^2 \\ \text{hypotenuse}^2=(2)/(4)\text{ + }(2)/(4)\text{ = }(4)/(4)=\text{ 1} \\ \text{hypotenuse = }\sqrt[]{1} \\ \text{hypotenuse = 1} \end{gathered}`

We can find csc θ, sin θ and cot θ now:

`\begin{gathered} co\sec \text{ }\theta\text{ = }\frac{1}{\sin \text{ }\theta} \\ \sin \text{ }\theta\text{ = }(opposite)/(hypotenuse) \\ \sin \theta\text{= }\frac{\frac{-\sqrt[]{2}}{2}}{1}\text{ = }\frac{-\sqrt[]{2}}{2} \\ \\ co\sec \text{ }\theta\text{ = }\frac{1}{\frac{-\sqrt[]{2}}{2}}\text{ = 1 }/\text{ }\frac{-\sqrt[]{2}}{2}\text{ = 1 }*\text{ }\frac{2}{-\sqrt[]{2}} \\ co\sec \text{ }\theta\text{ = }\frac{2}{-\text{ }\sqrt[]{2}} \end{gathered}`
`\begin{gathered} \sin \theta\text{ = }\frac{opposite}{\text{hypotenuse}} \\ \sin \theta\text{ = }\frac{-\text{ }\sqrt[]{2}}{2} \end{gathered}`
`\begin{gathered} \cot \theta\text{ = }\frac{1}{\tan\theta\text{ }} \\ \tan \text{ }\theta\text{ = }(\sin\theta)/(\cos\theta) \\ \\ \cos \text{ }\theta\text{ = }\frac{adjacent}{\text{hypotenuse}} \\ \cos \text{ }\theta\text{ = }\frac{\frac{-\sqrt[]{2}}{2}}{1}\text{ = }\frac{-\sqrt[]{2}}{2} \\ sin\text{ }\theta\text{ = }\frac{-\sqrt[]{2}}{2} \\ \\ \tan \text{ }\theta\text{ = }\frac{\frac{-\sqrt[]{2}}{2}}{\frac{-\sqrt[]{2}}{2}}\text{ = 1} \\ \cot \theta=(1)/(1) \\ \cot \theta\text{ = 1} \end{gathered}`