Let's first define the 6 trigonometric functions for a better understanding (here, O is "opposite," A is "adjacent," and H is "hypotenuse"):
[tex]sin(x) = \frac{O}{H} [/tex]
[tex]cos(x) = \frac{A}{H} [/tex]
[tex]tan(x) = \frac{O}{A} [/tex]
[tex]csc(x) = \frac{H}{O} [/tex]
[tex]sin(x) = \frac{H}{A} [/tex]
[tex]sin(x) = \frac{A}{O} [/tex]
Now, all we need to do is identify the opposite, and adjacent to the angle and solve for the hypotenuse based on given information. Let's first solve for hypotenuse:
[tex]c^2 = a^2 + b^2[/tex]
[tex]c = \sqrt{a^2+b^2} [/tex]
[tex]c = \sqrt{7^2+4^2} = \sqrt{65} [/tex]
We have all the information we need:
[tex]H = \sqrt{65}[/tex]
[tex]O = 7 [/tex]
[tex]A = 4 [/tex]
We can now solve for each ratio:
[tex]sin(x) = \frac{7}{ \sqrt{65} } = \frac{7 \sqrt{65} }{65} [/tex]
[tex]cos(x) = \frac{4}{ \sqrt{65} } = \frac{4 \sqrt{65} }{65} [/tex]
[tex]tan(x) = \frac{7}{ 4 } [/tex]
[tex]csc(x) = \frac{ \sqrt{65} }{7} [/tex]
[tex]sec(x) = \frac{ \sqrt{65} }{4} [/tex]
[tex]cot(x) = \frac{4}{7} [/tex]
Hope this helped!
