Question

current attempt in progress
please view the following video before answering this question. lecture video, section 4.3, example 02
point (-3,4) is on the terminal side of angle θ. find all six trigonometric functions of θ. give exact answers.
sinθ=
cosθ=
tanθ=
cscθ=
secθ=
cotθ=

Explanation:

Step1: Calculate the value of \(r\)

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

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{y}{r}\), substituting \(y = 4\) and \(r = 5\), we get \(\sin\theta=\frac{4}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{x}{r}\), substituting \(x=-3\) and \(r = 5\), we get \(\cos\theta=-\frac{3}{5}\)

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{y}{x}\), substituting \(x=-3\) and \(y = 4\), we get \(\tan\theta=-\frac{4}{3}\)

Step5: Calculate \(\csc\theta\)

\(\csc\theta=\frac{r}{y}\), substituting \(y = 4\) and \(r = 5\), we get \(\csc\theta=\frac{5}{4}\)

Step6: Calculate \(\sec\theta\)

\(\sec\theta=\frac{r}{x}\), substituting \(x=-3\) and \(r = 5\), we get \(\sec\theta=-\frac{5}{3}\)

Step7: Calculate \(\cot\theta\)

\(\cot\theta=\frac{x}{y}\), substituting \(x=-3\) and \(y = 4\), we get \(\cot\theta=-\frac{3}{4}\)

Answer:

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