Question

let \\( \theta \\) be the angle in standard position whose terminal side contains the given point. compute \\( \cos ( \theta ) \\) and \\( \sin ( \theta ) \\). answer exactly. \
\
\\( ( 2,1 ) \\)\
\
\\( \sin ( \theta ) = \\)\
\\( \cos ( \theta ) = \\)\
\
question help: \\( \square \\) message instructor

Explanation:

Step1: Calculate the radius \( r \)

For a point \((x,y)=(2,1)\), use the formula \( r=\sqrt{x^{2}+y^{2}} \).

$$ r = \sqrt{2^{2}+1^{2}}=\sqrt{4 + 1}=\sqrt{5} $$

Step2: Compute \(\sin(\theta)\) and \(\cos(\theta)\)

Use the definitions \(\sin(\theta)=\frac{y}{r}\) and \(\cos(\theta)=\frac{x}{r}\).
For \(\sin(\theta)\):

$$ \sin(\theta)=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5} $$

For \(\cos(\theta)\):

$$ \cos(\theta)=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5} $$

Answer:

\(\sin(\theta)=\frac{\sqrt{5}}{5}\), \(\cos(\theta)=\frac{2\sqrt{5}}{5}\)