Question

The angle 01 is located in Quadrant III, and sin(01)=-12/13. What is the value of cos(O1)

Answer 1

$cos\emptyset_1=\frac{5}{\text{1}3}$

Explanation

Step 1

The angle 01 is located in Quadrant III,so

we know, by definition

$\text{sen}\varphi=\frac{opposite\text{ side}}{\text{hypotenuse}}$

so, by replacing

$\begin{gathered} \text{sen}\varphi=\frac{opposite\text{ side}}{\text{hypotenuse}} \\ \text{sen}\emptyset_1=\frac{-12}{\text{1}3} \\ \text{hence:} \\ \text{opposite side}(\text{purple) is 12( negative indicates to the left)} \\ \text{hypotenuse}=13 \end{gathered}$

to find cos , we need to find the misssing side( adjacent side), let's use the Pythagorean theorem:

$\begin{gathered} a^2+b^2=c^2 \\ so \\ 12^2+adjacentside^2=13^2 \\ adjacentside^2=13^2-12^2 \\ adjacentside^2=25 \\ \sqrt{adjacentside^2}=\sqrt{25} \\ \text{adjacent side= 5} \\ \\ \end{gathered}$

Step 2

now, replace in the cosine formula

$\begin{gathered} cos\emptyset=\frac{adjancent\text{ side}}{\text{hypotenuse}} \\ cos\emptyset_1=\frac{5}{\text{1}3} \\ \end{gathered}$

I hope this helps you