Question

Find sin θ and cos θ where θ is the angle that corresponds to the point P (5,12)

Answer 1

Check the picture below.

`\bf P~(\stackrel{a}{5}~~,~~\stackrel{b}{12})\qquad \impliedby \textit{let's find the hypotenuse} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=√(a^2+b^2) \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases}`

`\bf c=√(5^2+12^2)\implies c=√(25+144)\implies c=√(169)\implies c=13 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill sin(\theta )=\cfrac{\stackrel{opposite}{12}}{\stackrel{hypotenuse}{13}}\qquad \qquad cos(\theta )=\cfrac{\stackrel{adjacent}{5}}{\stackrel{hypotenuse}{13}}~\hfill`

Answer 2

The answer to your question is:

sin Ф = 12/13

cos Ф = 5/13

Explanation:

Data

Point (5,12)

sin Ф = ?

cos Ф = ?

Process

1.- Plot the point, and draw a right triangle from the points (0, 0), (5, 0) and (5, 12). See the picture below.

2.- Calculate the hypotenuse.

3.- Calculate sin Ф and cos Ф.

c² = 5² + 12²

c² = 25 + 144

c² = 169

c = 13

sin Ф = opposite side/ hypotenuse = 12/13

cos Ф = adjacent side/ hypotenuse = 5/13