Question

current attempt in progress
please view the following video before answering this question. lecture video, section 4.3, example 02
point (4, - 3) is on the terminal side of angle θ. find all six trigonometric functions of θ. give exact answers.
$\sin\theta =$
$\cos\theta =$
$\tan\theta =$
$\csc\theta =$
$\sec\theta =$
$\cot\theta =$

Explanation:

Step1: Find the value of \(r\)

For a point \((x,y)\) on the terminal side of an angle \(\theta\), \(r=\sqrt{x^{2}+y^{2}}\). Given \(x = 4\) and \(y=-3\), then \(r=\sqrt{4^{2}+(-3)^{2}}=\sqrt{16 + 9}=\sqrt{25}=5\).

Step2: Calculate \(\sin\theta\)

By the definition \(\sin\theta=\frac{y}{r}\), substituting \(y=-3\) and \(r = 5\), we get \(\sin\theta=\frac{-3}{5}\).

Step3: Calculate \(\cos\theta\)

Using the definition \(\cos\theta=\frac{x}{r}\), with \(x = 4\) and \(r = 5\), we have \(\cos\theta=\frac{4}{5}\).

Step4: Calculate \(\tan\theta\)

According to the formula \(\tan\theta=\frac{y}{x}\), substituting \(x = 4\) and \(y=-3\), we obtain \(\tan\theta=\frac{-3}{4}\).

Step5: Calculate \(\csc\theta\)

Since \(\csc\theta=\frac{r}{y}\), with \(r = 5\) and \(y=-3\), we get \(\csc\theta=\frac{5}{-3}=-\frac{5}{3}\).

Step6: Calculate \(\sec\theta\)

Using \(\sec\theta=\frac{r}{x}\), substituting \(r = 5\) and \(x = 4\), we have \(\sec\theta=\frac{5}{4}\).

Step7: Calculate \(\cot\theta\)

According to \(\cot\theta=\frac{x}{y}\), with \(x = 4\) and \(y=-3\), we obtain \(\cot\theta=\frac{4}{-3}=-\frac{4}{3}\).

Answer:

\(\sin\theta=-\frac{3}{5}\), \(\cos\theta=\frac{4}{5}\), \(\tan\theta=-\frac{3}{4}\), \(\csc\theta=-\frac{5}{3}\), \(\sec\theta=\frac{5}{4}\), \(\cot\theta=-\frac{4}{3}\)