Step 1: Use the Pythagorean Theorem to find the missing length: a² + b² = c²
[tex]5^2+7^2=c^2\\25+49=c^2\\74=c^2\\\sqrt{74}=c[/tex]
from the perspective of X:
- opposite = 5
- adjacent = 7
- hypotenuse = √74
from the perspective of Y:
- opposite = 7
- adjacent = 5
- hypotenuse = √74
[tex]sin\ x=\dfrac{opposite}{hypotenuse}=\dfrac{5}{\sqrt{74}}=\dfrac{5\sqrt{74}}{74}\\\\csc\ y=\dfrac{hypotenuse}{opposite}=\dfrac{\sqrt{74}}{7}\\\\tan\ x=\dfrac{opposite}{adjacent}=\dfrac{5}{7}\\\\cot\ y=\dfrac{adjacent}{opposite}=\dfrac{5}{7}\\\\cos\ x=\dfrac{adjacent}{hypotenuse}=\dfrac{7}{\sqrt{74}}=\dfrac{7\sqrt{74}}{74}\\\\sec\ y=\dfrac{hypotenuse}{adjacent}=\dfrac{\sqrt{74}}{5}[/tex]
