Question

let \\( \theta \\) be the angle in standard position whose terminal side contains the given point. compute \\( \cos ( \theta ) \\) and \\( \sin ( \theta ) \\). answer exactly.

\\( ( - 5,12 ) \\)

\\( \sin ( \theta ) = \\)
\\( \cos ( \theta ) = \\)

question help: video message instructor

Explanation:

Step1: Calculate the radius \( r \)

For a point \((x,y)\) in the coordinate plane, \( r=\sqrt{x^{2}+y^{2}}\). Here \(x = - 5\) and \(y = 12\), so \(r=\sqrt{(-5)^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\).

Step2: Calculate \(\sin(\theta)\)

The formula for \(\sin(\theta)\) is \(\sin(\theta)=\frac{y}{r}\). Substituting \(y = 12\) and \(r = 13\), we get \(\sin(\theta)=\frac{12}{13}\).

Step3: Calculate \(\cos(\theta)\)

The formula for \(\cos(\theta)\) is \(\cos(\theta)=\frac{x}{r}\). Substituting \(x=-5\) and \(r = 13\), we get \(\cos(\theta)=-\frac{5}{13}\).

Answer:

\(\sin(\theta)=\frac{12}{13}\), \(\cos(\theta)=-\frac{5}{13}\)