Chapter 14.3, Problem 32PPSE

Solution

To find: the values of all trigonometric functions of $\theta$ .

  $\cos \theta =\sqrt{0.8775},\tan \theta =\frac{0.35\sqrt{0.8775}}{0.8775},\csc \theta =\frac{1}{0.35},\sec \theta =\frac{\sqrt{0.8775}}{0.8775},\cot \theta =\frac{\sqrt{0.8775}}{0.35}$ .

Given:

  $\sin \theta =0.35$ .

Concept used:

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.

Calculation:

Let us draw the right triangle based on the given trigonometric ratio as follows:

Since the sine function is the ratio of the opposite side to the hypotenuse, therefore the triangle is,

  

Now by using the Pythagorean Theorem in the triangle ABC, it follows that

  $\begin{array}{c}{\left(\text{AC}\right)}^2={\left(\text{AB}\right)}^2+{\left(\text{BC}\right)}^2\\ 1^2={\left(0.35\right)}^2+{\left(\text{BC}\right)}^2\\ 1=0.1225+{\left(\text{BC}\right)}^2\\ {\left(\text{BC}\right)}^2=0.8775\\ \text{BC}=\sqrt{0.8775}\end{array}$

Now,

  $\begin{array}{c}\cos \theta =\frac{\text{BC}}{\text{AC}}\\ =\frac{\sqrt{0.8775}}{1}\\ =\sqrt{0.8775}\end{array}$

  $\begin{array}{c}\tan \theta =\frac{\text{AB}}{\text{BC}}\\ =\frac{0.35}{\sqrt{0.8775}}\\ =\frac{0.35\sqrt{0.8775}}{0.8775}\end{array}$

  $\begin{array}{c}\csc \theta =\frac{\text{AC}}{\text{AB}}\\ =\frac{1}{0.35}\end{array}$

  $\begin{array}{c}\sec \theta =\frac{\text{AC}}{\text{BC}}\\ =\frac{1}{\sqrt{0.8775}}\\ =\frac{\sqrt{0.8775}}{0.8775}\end{array}$

And,

  $\begin{array}{c}\cot \theta =\frac{\text{BC}}{\text{AB}}\\ =\frac{\sqrt{0.8775}}{0.35}\end{array}$