Chapter C, Problem 13E

To determine

To calculate:

The remaining trigonometric ratios:

  $\sin \theta =\frac{3}{5},0<\theta <\frac{\pi }{2}$

Answer to Problem 13E

The remaining trigonometric ratios are listed as:

  $\begin{array}{l}\cos ec\theta =\frac{r}{y}=\frac{5}{3}\\ \cos \theta =\frac{x}{r}=\frac{5}{4}\\ \sec \theta =\frac{r}{x}=\frac{4}{5}\\ \tan \theta =\frac{y}{x}=\frac{3}{4}\\ \cot \theta =\frac{x}{y}=\frac{4}{3}\end{array}$

Explanation of Solution

Given information:

  $\sin \theta =\frac{3}{5},0<\theta <\frac{\pi }{2}$

Formula Used:

Trigonometric ratios are given by the following formula:

  $\begin{array}{l}\sin \theta =\frac{y}{r}\cos ec\theta =\frac{r}{y}\\ \cos \theta =\frac{x}{r}\sec \theta =\frac{r}{x}\\ \tan \theta =\frac{y}{x}\cot \theta =\frac{x}{y}\end{array}$

Calculation:

From the given data:

  $\begin{array}{l}\sin \theta =\frac{x}{r}\\ \sin \theta =\frac{3}{5}\end{array}$

So, $x=3,r=5$ the angle is constructed as below:

  

Now, find the value of $y$

  $\begin{array}{l}{r}^2=x^2+y^2\\ {\left(5\right)}^2={\left(3\right)}^2+y^2\\ y^2=25-9\\ y^2=16\\ y=4\end{array}$

Therefore, remaining trigonometric ratios are calculated as:

  $\begin{array}{l}\cos ec\theta =\frac{r}{y}=\frac{5}{3}\\ \cos \theta =\frac{x}{r}=\frac{5}{4}\\ \sec \theta =\frac{r}{x}=\frac{4}{5}\\ \tan \theta =\frac{y}{x}=\frac{3}{4}\\ \cot \theta =\frac{x}{y}=\frac{4}{3}\end{array}$