Chapter 13: Q18. (page 706)

Find the values of six trigonometric functions for angle $(θ)$ shown below.

Short Answer

Expert verified

The six trigonometric functions are: $\sin θ = \frac{9}{\sqrt{106}}, \csc θ = \frac{\sqrt{106}}{9}, \cos θ = \frac{5}{\sqrt{106}}, \sec θ = \frac{\sqrt{106}}{5}, \tan θ = \frac{9}{5}, \cot θ = \frac{5}{9}$ .

Step by step solution

Step 1. Write down the given information.

The given figure is shown below.

Step 2. Concept used.

The six trigonometric functions are $\sin θ, \csc θ, \cos θ, \sec θ, \tan θ, \cot θ$ .

Let $(x, y)$ be a point in X-Y plane and (r)be the length of a line segment drawn from the origin to the point.

In, $Δ O A B, O A = r, O B = x, A B = y$

Therefore, six trigonometric functions are;

$\begin{matrix} \sin θ = \frac{A B}{O A} = \frac{y}{r}, \csc θ = \frac{1}{\sin θ} = \frac{O A}{A B} = \frac{r}{y} \\ \cos θ = \frac{O B}{O A} = \frac{x}{r}, \sec θ = \frac{1}{\cos θ} = \frac{O A}{O B} = \frac{r}{x} \\ \tan θ = \frac{A B}{O B} = \frac{y}{x}, \cot θ = \frac{1}{\tan θ} = \frac{O B}{A B} = \frac{x}{y} \end{matrix}$

Step 3. Calculation.

From the given figure, the hypotenuse is evaluated as:

$\begin{matrix} Hypotenuse = \sqrt{5^{2} + 9^{2}} \\ = \sqrt{106} \end{matrix}$

Now, the six trigonometric functions are evaluated as:

$\begin{matrix} \sin θ = \frac{9}{\sqrt{106}} \csc θ = \frac{\sqrt{106}}{9} \\ \cos θ = \frac{5}{\sqrt{106}} \sec θ = \frac{\sqrt{106}}{5} \\ \tan θ = \frac{9}{5} \cot θ = \frac{5}{9} \end{matrix}$

Step 4. Conclusion.

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Find the values of six trigonometric functions for angle $(θ)$ shown below.

Short Answer

Step by step solution

Step 1. Write down the given information.

Step 2. Concept used.

Step 3. Calculation.

Step 4. Conclusion.

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Find the values of six trigonometric functions for angle θ shown below.

Short Answer

Step by step solution

Step 1. Write down the given information.

Step 2. Concept used.

Step 3. Calculation.

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Calculus

Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the values of six trigonometric functions for angle $(θ)$ shown below.