Problem 25 - Basic Technical Mathematics with Calculus

Question

Each given point is on the terminal side of an angle. Show that each of the given ficnctions is the same for each points. \((3,4),(6,8),(4.5,6), \sin \theta\) and \(\tan \theta.\)

Step-by-Step Solution

Answer

Both \(\sin \theta\) and \(\tan \theta\) are the same for all points: \(\sin \theta = \frac{4}{5}\) and \(\tan \theta = \frac{4}{3}\).

Step 1: Understanding the Question

We need to verify if the values of \( \sin \theta \) and \( \tan \theta \) are the same for the points \((3,4)\), \((6,8)\), and \((4.5,6)\). Each point lies on the terminal side of an angle in standard position.

Step 2: Calculate the Hypotenuse

For each point \((x, y)\), calculate the hypotenuse \(r\) of the right triangle formed, using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \]

Step 3: Calculate \(\sin \theta\) for Each Point

\( \sin \theta = \frac{y}{r} \). Calculate \(\sin \theta\) for each point:- For \((3,4)\), \( r = \sqrt{3^2 + 4^2} = 5 \), so \( \sin \theta = \frac{4}{5}\).- For \((6,8)\), \( r = \sqrt{6^2 + 8^2} = 10 \), so \( \sin \theta = \frac{8}{10} = \frac{4}{5}\).- For \((4.5,6)\), \( r = \sqrt{4.5^2 + 6^2} = 7.5 \), so \( \sin \theta = \frac{6}{7.5} = \frac{4}{5}\).

Step 4: Calculate \(\tan \theta\) for Each Point

\( \tan \theta = \frac{y}{x} \). Calculate \(\tan \theta\) for each point:- For \((3,4)\), \( \tan \theta = \frac{4}{3} \).- For \((6,8)\), \( \tan \theta = \frac{8}{6} = \frac{4}{3} \).- For \((4.5,6)\), \( \tan \theta = \frac{6}{4.5} = \frac{4}{3} \).

Step 5: Conclusion

Since \(\sin \theta = \frac{4}{5}\) and \(\tan \theta = \frac{4}{3}\) for each point, both \(\sin \theta\) and \(\tan \theta\) are the same for all the given points.

Key Concepts

Understanding Right Triangles

A right triangle is a geometric figure with one of its angles exactly equal to 90 degrees. In a right triangle, the side opposite the right angle is the longest, known as the hypotenuse. The other two sides are called the legs of the triangle.

Right triangles have special properties that allow us to calculate unknown side lengths and angles using trigonometric functions. This is especially helpful in many areas such as architecture, navigation, and when solving problems involving these geometric shapes.

One of the key aspects of a right triangle is knowing how to identify these elements:

• The base or adjacent side: next to the angle of interest.
• The opposite side: across from the angle of interest.
• The hypotenuse: always opposite the right angle.

Understanding these components aids in correctly applying trigonometric formulas to solve for missing values or angles.

Exploring the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially concerning right triangles. It states that the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). The theorem can be expressed by the formula: \[ a^2 + b^2 = c^2 \]

This theorem is instrumental in calculating the hypotenuse or other sides of a right triangle when the lengths of two sides are known.

In the exercise involving points \((3,4), (6,8), (4.5,6)\)—the hypotenuse, \(r\), is calculated using this theorem:

• For point \((3,4)\), \(r = \sqrt{3^2 + 4^2} = 5\)
• For point \((6,8)\), \(r = \sqrt{6^2 + 8^2} = 10\)
• For point \((4.5,6)\), \(r = \sqrt{4.5^2 + 6^2} = 7.5\)

These calculations confirm each triangle's hypotenuse, aiding in determining trigonometric ratios.

The Basics of the Sine Function

The sine function, a key component in trigonometry, relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is given by the formula:\(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\)

In the exercise, for each point given on the terminal side of an angle, the sine function is calculated as follows:

• For point \((3,4)\), \(\sin \theta = \frac{4}{5}\)
• For point \((6,8)\), \(\sin \theta = \frac{8}{10} = \frac{4}{5}\)
• For point \((4.5,6)\), \(\sin \theta = \frac{6}{7.5} = \frac{4}{5}\)

With \(\sin \theta\) consistently equal to \(\frac{4}{5}\) for each coordinate pair, we see how the sine function remains constant across different points, assuming they form similar right triangles.

Understanding the Tangent Function

The tangent function is another pivotal trigonometric concept that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. The formula for calculating the tangent of an angle is:\(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\)

In our exercise, let's see how \(\tan \theta\) is calculated for each point:

• For point \((3,4)\), \(\tan \theta = \frac{4}{3}\)
• For point \((6,8)\), \(\tan \theta = \frac{8}{6} = \frac{4}{3}\)
• For point \((4.5,6)\), \(\tan \theta = \frac{6}{4.5} = \frac{4}{3}\)

We observe that \(\tan \theta\) remains consistent as \(\frac{4}{3}\) across the mentioned points. This reflects the similarity in angles formed despite differing triangle sizes, owing to proportional side lengths.