Problem 7 - 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. $$(0.09,0.40)$$

Step-by-Step Solution

Answer

\(\sin(\theta) = \frac{0.40}{0.41}\), \(\cos(\theta) = \frac{0.09}{0.41}\), \(\tan(\theta) = \frac{0.40}{0.09}\).

Step 1: Understand the Problem

We know the angle's terminal side passes through the point \((0.09, 0.40)\). We will use this point to find the values of the trigonometric functions. To do this, we first need to find the radius \(r\), which is the distance from the origin \((0,0)\) to the point \((0.09, 0.40)\).

Step 2: Find the Radius

The radius \(r\) can be found using the distance formula: \(r = \sqrt{(x^2 + y^2)}\). Substitute the given values \((x, y) = (0.09, 0.40)\): \[ r = \sqrt{(0.09)^2 + (0.40)^2} = \sqrt{0.0081 + 0.16} = \sqrt{0.1681} = 0.41 \]

Step 3: Calculate Sine

The sine of the angle \(\theta\) is given by \(\sin(\theta) = \frac{y}{r}\). Substitute \(y = 0.40\) and \(r = 0.41\): \[ \sin(\theta) = \frac{0.40}{0.41} \]

Step 4: Calculate Cosine

The cosine of the angle \(\theta\) is \(\cos(\theta) = \frac{x}{r}\). Substitute \(x = 0.09\) and \(r = 0.41\): \[ \cos(\theta) = \frac{0.09}{0.41} \]

Step 5: Calculate Tangent

The tangent of the angle \(\theta\) is \(\tan(\theta) = \frac{y}{x}\). Substitute \(y = 0.40\) and \(x = 0.09\): \[ \tan(\theta) = \frac{0.40}{0.09} \]

Key Concepts

Sine

The sine function plays a crucial role in trigonometry and is one of the primary functions used to describe the relationships in a right triangle and on the unit circle. This function specifically relates to the vertical side opposite the angle being considered.
Let’s imagine a right triangle where one angle is \( \theta \) and the triangle is placed with its vertex at the origin in the Cartesian plane. The point \((0.09, 0.40)\) lies on the terminal side of this angle.
In trigonometric terms, the sine of angle \(\theta\) is determined by the ratio of the length of the side opposite the angle to the hypotenuse of the triangle.

• Here, the opposite side's length is the 'y' coordinate, which is 0.40.
• The hypotenuse, or radius \(r\), calculated using the distance formula, is 0.41.
• Therefore, \(\sin(\theta) = \frac{0.40}{0.41}\).

This means for our specific scenario with point \((0.09, 0.40)\), sine captures how high the point is above the X-axis relative to the origin.

Cosine

The cosine function is another pillar of trigonometric functions and is used extensively in various mathematical and physical applications. It describes how far along the horizontal axis a point is from the origin.
When examining the right triangle with the terminal side of angle \(\theta\) going through the point \((0.09, 0.40)\), the cosine is defined as the adjacent side's length over the hypotenuse.

• The adjacent side relates to the 'x' coordinate, which is 0.09.
• Again, our hypotenuse, or radius, is 0.41.
• Thus, \(\cos(\theta) = \frac{0.09}{0.41}\).

Cosine provides insight into how far the point stretches along the X-axis, reflecting the horizontal component of movement from the origin.

Tangent

The tangent function is unique compared to sine and cosine and is calculated through the ratio of the vertical change to the horizontal change for the angle \(\theta\). It’s particularly useful for understanding the slope of the line formed by the angle.
In our right triangle setup, tangent is represented by the ratio of the length of the opposite side to the adjacent side.

• The opposite side, associated with the 'y' coordinate, measures 0.40.
• The adjacent side, linked to the 'x' coordinate, is 0.09.
• Therefore, \(\tan(\theta) = \frac{0.40}{0.09}\).

A tangent's value reflects how steeply the point at \((0.09, 0.40)\) rises compared to its horizontal spread. This concept can be visualized as the slope formed by the line intersecting the angle's terminal side and shortened axis extensions.