The point (-2,7) defines an angle 𝜃 in standard position. Find sin 𝜃.
Provide a sketch and calculations to support your answer. Write the ratio in reduced fractional
form with a rationalized denominator if necessary.
To find sin 𝜃, we first need to determine the length of the opposite side (y-coordinate) and the hypotenuse of the right triangle formed by the point (-2,7) in the standard position.
The opposite side is the y-coordinate of the point, which is 7.
The hypotenuse can be found using the Pythagorean theorem:
hypotenuse = √((-2)^2 + 7^2)
= √(4 + 49)
= √53
Now, sin 𝜃 is calculated as:
sin 𝜃 = opposite/hypotenuse
= 7/√53
Thus, sin 𝜃 in reduced fractional form with a rationalized denominator is 7√53 / 53.
Below is a sketch of the right triangle formed by the point (-2,7) in the standard position:
```
|\
| \
| \ 7
| \
|____\
2
```