Chapter 14, Problem 40MCQ

Solution

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

  $\sin \theta =\frac{11\sqrt{137}}{137},\cos \theta =\frac{4\sqrt{137}}{137},\csc \theta =\frac{\sqrt{137}}{11},\sec \theta =\frac{\sqrt{137}}{4},\cot \theta =\frac{4}{11}$ .

Given:

  $\tan \theta =\frac{11}{4}$ .

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

  

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

  $\begin{array}{c}{\left(\text{AC}\right)}^2={\left(\text{AB}\right)}^2+{\left(\text{BC}\right)}^2\\ {\left(\text{AC}\right)}^2={\text{11}}^2+{\text{4}}^2\\ {\left(\text{AC}\right)}^2=121+16\\ {\left(\text{AC}\right)}^2=137\\ \text{AC}=\sqrt{137}\end{array}$

Now,

  $\begin{array}{c}\sin \theta =\frac{\text{AB}}{\text{AC}}\\ =\frac{11}{\sqrt{137}}\\ =\frac{11\sqrt{137}}{137}\end{array}$

  $\begin{array}{c}\cos \theta =\frac{\text{BC}}{\text{AC}}\\ =\frac{4}{\sqrt{137}}\\ =\frac{4\sqrt{137}}{137}\end{array}$

  $\begin{array}{c}\csc \theta =\frac{\text{AC}}{\text{AB}}\\ =\frac{\sqrt{137}}{11}\end{array}$

  $\begin{array}{c}\sec \theta =\frac{\text{AC}}{\text{BC}}\\ =\frac{\sqrt{137}}{4}\end{array}$

And,

  $\begin{array}{c}\cot \theta =\frac{\text{BC}}{\text{AB}}\\ =\frac{4}{11}\end{array}$