Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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**Title: Finding the Tangent of an Angle α**

**Objective:**
Learn how to find the tangent of an angle given a point on the coordinate plane.

**Concept Overview:**

In this diagram, we are tasked with finding the tangent of the angle α, which is formed by a vector \( r \) originating from the origin and terminating at the point \((\sqrt{5}, -\sqrt{7})\).

**Diagram Explanation:**

- A coordinate plane is depicted with both horizontal and vertical axes.
- The vector \( r \) makes an angle α with the horizontal axis.
- The coordinates of the endpoint of the vector are \((\sqrt{5}, -\sqrt{7})\).

**Steps to Find \(\tan \alpha\):**

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:

\[
\tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}}
\]

For a point \((x, y)\),

- \(\tan \alpha = \frac{y}{x}\)

Plugging in the coordinates:

- \(\tan \alpha = \frac{-\sqrt{7}}{\sqrt{5}}\)

**Conclusion:**

The tangent of angle α is \(\frac{-\sqrt{7}}{\sqrt{5}}\), which simplifies the trigonometric analysis of the vector in relation to the coordinate plane. Understanding this ratio is fundamental in relating algebraic expressions to geometric representations in trigonometry.
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Transcribed Image Text:**Title: Finding the Tangent of an Angle α** **Objective:** Learn how to find the tangent of an angle given a point on the coordinate plane. **Concept Overview:** In this diagram, we are tasked with finding the tangent of the angle α, which is formed by a vector \( r \) originating from the origin and terminating at the point \((\sqrt{5}, -\sqrt{7})\). **Diagram Explanation:** - A coordinate plane is depicted with both horizontal and vertical axes. - The vector \( r \) makes an angle α with the horizontal axis. - The coordinates of the endpoint of the vector are \((\sqrt{5}, -\sqrt{7})\). **Steps to Find \(\tan \alpha\):** The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: \[ \tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}} \] For a point \((x, y)\), - \(\tan \alpha = \frac{y}{x}\) Plugging in the coordinates: - \(\tan \alpha = \frac{-\sqrt{7}}{\sqrt{5}}\) **Conclusion:** The tangent of angle α is \(\frac{-\sqrt{7}}{\sqrt{5}}\), which simplifies the trigonometric analysis of the vector in relation to the coordinate plane. Understanding this ratio is fundamental in relating algebraic expressions to geometric representations in trigonometry.
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