Question

1. draw a diagram and find the exact, simplified values given that $tan\beta = 3$ and the terminal side of $\beta$ is in quadrant i.

$sin\beta =$
$cot\beta =$

Explanation:

Step1: Recall the definition of tangent

Since \(\tan\beta=\frac{y}{x} = 3=\frac{3}{1}\), we can assume \(y = 3\) and \(x=1\) (in the first - quadrant where \(x>0,y>0\)).

Step2: Use the Pythagorean theorem to find \(r\)

By the formula \(r=\sqrt{x^{2}+y^{2}}\), substituting \(x = 1\) and \(y = 3\), we get \(r=\sqrt{1^{2}+3^{2}}=\sqrt{1 + 9}=\sqrt{10}\).

Step3: Calculate \(\sin\beta\)

Using the formula \(\sin\beta=\frac{y}{r}\), substituting \(y = 3\) and \(r=\sqrt{10}\), we have \(\sin\beta=\frac{3}{\sqrt{10}}=\frac{3\sqrt{10}}{10}\).

Step4: Calculate \(\cot\beta\)

Using the formula \(\cot\beta=\frac{x}{y}\), substituting \(x = 1\) and \(y = 3\), we get \(\cot\beta=\frac{1}{3}\).

Answer:

\(\sin\beta=\frac{3\sqrt{10}}{10}\), \(\cot\beta=\frac{1}{3}\)