Problem 13 - Basic Technical Mathematics with Calculus

Question

Find values of the trigonometric functions of the angle (in standard position) whose terminal side passes through the given points. For Exercises \(3-14,\) give answers in exact form. For Exercises 15 and \(16,\) the coordinates are approximate. $$(50,20)$$

Step-by-Step Solution

Answer

Sine: \(\frac{2\sqrt{29}}{29}\), Cosine: \(\frac{5\sqrt{29}}{29}\), Tangent: \(\frac{2}{5}\).

Step 1: Calculate the hypotenuse

First, we need to find the hypotenuse of the right triangle formed by the point \((50, 20)\). The hypotenuse is the distance from the origin \((0, 0)\) to the point \((50, 20)\). We will use the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{50^2 + 20^2} = \sqrt{2500 + 400} = \sqrt{2900} = 10\sqrt{29} \]

Step 2: Find the sine of the angle

The sine function is defined as the ratio of the opposite side to the hypotenuse. Here, the opposite side is the y-coordinate of the point, which is 20. So, \[ \sin(\theta) = \frac{20}{10\sqrt{29}} = \frac{2}{\sqrt{29}} \] To express this in exact form, multiply the numerator and the denominator by \(\sqrt{29}\): \[ \sin(\theta) = \frac{2\sqrt{29}}{29} \]

Step 3: Calculate the cosine of the angle

The cosine function is the ratio of the adjacent side (x-coordinate) to the hypotenuse. Here, the adjacent side is 50. So, \[ \cos(\theta) = \frac{50}{10\sqrt{29}} = \frac{5}{\sqrt{29}} \] Rationalizing the denominator: \[ \cos(\theta) = \frac{5\sqrt{29}}{29} \]

Step 4: Determine the tangent of the angle

The tangent function is the ratio of the opposite side to the adjacent side. Thus, \[ \tan(\theta) = \frac{y}{x} = \frac{20}{50} = \frac{2}{5} \]

Key Concepts

Understanding Right Triangles

A right triangle is a type of triangle that features one angle equal to 90 degrees, commonly referred to as a right angle. This fundamental geometry form allows us to explore relationships between its sides and angles through trigonometry. The right angle forms at the intersection of two perpendicular sides, known as the 'legs' of the triangle - these are the base and the height.

• One of the legs opposes the right angle and is named the hypotenuse. It is the longest side of the triangle.
• The other two sides - adjacent and opposite - correspond to the specific angle we are working with.

In our case, the triangle is formed by placing the vertex at the origin (0,0) of a coordinate system, thus simplifying calculations with the distances representing sides.

Using the Distance Formula

The distance formula is a crucial tool in coordinate geometry. It helps calculate the length of the hypotenuse when we know the coordinates of the point that forms a right triangle with the origin.

• The distance formula states: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• For the origin and a point \((x, y)\), this simplifies to: \[ r = \sqrt{x^2 + y^2} \]

Applying this to our given point (50, 20), we find the hypotenuse as \[ r = \sqrt{50^2 + 20^2} = 10\sqrt{29} \] This result is fundamental in calculating all the trigonometric functions.

Exploring the Sine Function

The sine function is defined in the context of a right triangle as the ratio between the length of the opposite side to the angle, and the hypotenuse.

• In mathematical terms: \[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]
• For our specific example, with an opposite side of 20 and a hypotenuse of \( 10\sqrt{29} \), it becomes: \[ \sin(\theta) = \frac{20}{10\sqrt{29}} = \frac{2\sqrt{29}}{29} \] after rationalizing the denominator.

Grasping this concept is integral for solving various trigonometric problems, especially when dealing with angles and coordinates.

Understanding the Cosine Function

The cosine function in right triangle trigonometry is associated with the ratio of the length of the adjacent side to the hypotenuse.

• It can be expressed as:\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
• Given our triangle, where the adjacent side is 50, the formula becomes:\[ \cos(\theta) = \frac{50}{10\sqrt{29}} \]
• Rationalizing the denominator leads to:\[ \cos(\theta) = \frac{5\sqrt{29}}{29} \]

This function is pivotal in numerous calculations involving angles and dimensions in physics, engineering, and beyond.

Delving into the Tangent Function

In right triangle trigonometry, the tangent function reflects the ratio of the opposite side to the adjacent side of the angle of interest.

• Mathematically, we understand it as:\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
• From our point, this ratio is notably straightforward:\[ \tan(\theta) = \frac{20}{50} = \frac{2}{5} \]

The tangent function aids in comprehending angular relationships and is indispensable in practical applications like navigation, surveying, and architecture.