Problem 8 - 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. $$(1.1,6.0)$$

Step-by-Step Solution

Answer

Since the coordinates are approximate, the exact trigonometric function values involve irrational numbers.

Step 1: Understand the Problem

We are given a point \((1.1, 6.0)\) through which the terminal side of an angle passes. The task is to find the trigonometric functions (sine, cosine, and tangent) of this angle in exact form.

Step 2: Use the Pythagorean Theorem

Calculate the radius (or hypotenuse) \(r\) of the right triangle formed by the point. The formula for \(r\) is \(r = \sqrt{x^2 + y^2}\). Substitute \(x = 1.1\) and \(y = 6.0\).

Step 3: Calculate the Hypotenuse

Compute \(r = \sqrt{(1.1)^2 + (6.0)^2} = \sqrt{1.21 + 36} = \sqrt{37.21}\).

Step 4: Express Trigonometric Functions

Using \(r = \sqrt{37.21}\), express the trigonometric functions. The sine of the angle is \( \sin(\theta) = \frac{y}{r} = \frac{6.0}{\sqrt{37.21}}\), the cosine is \( \cos(\theta) = \frac{x}{r} = \frac{1.1}{\sqrt{37.21}}\), and the tangent is \( \tan(\theta) = \frac{y}{x} = \frac{6.0}{1.1}\).

Step 5: Simplify and Approximate

Though exact answers are required, note that the denominators involve the square root of a decimal, which makes them not simplify to exact integer values. To simplify: - \( \sin(\theta) \approx \frac{6.0}{6.1} \approx 0.9836\), - \( \cos(\theta) \approx \frac{1.1}{6.1} \approx 0.1803 \), - \( \tan(\theta) \approx 5.4545 \).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is an essential concept in trigonometry, especially when dealing with right triangles. It relates the lengths of the sides of a right triangle—specifically, it states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as:

• \( c^2 = a^2 + b^2 \)

This theorem is crucial when you need to find the hypotenuse given the other two sides. In the exercise, the point

• \((x, y) = (1.1, 6.0)\)

can be thought of as the end of the two legs of a right triangle with the origin. The hypotenuse \(r\) is found using:

• \( r = \sqrt{x^2 + y^2} = \sqrt{1.1^2 + 6.0^2} \)

This allows finding \(r\) to apply it in trigonometric functions of the angle.

exact form

"Exact form" requires expressing numbers with complete precision, without decimal approximations. This often involves leaving numbers in terms like fractions, square roots, or constants instead of converting them into decimal form. In our exercise, the trigonometric functions need to be in exact form. For instance:

• \( \sin(\theta) = \frac{6.0}{\sqrt{37.21}} \)
• \( \cos(\theta) = \frac{1.1}{\sqrt{37.21}} \)
• \( \tan(\theta) = \frac{6.0}{1.1} \)

Each of these expressions uses the calculated hypotenuse \(r\) from the Pythagorean theorem. Although simpler decimal forms can often be more intuitive, keeping the exact roots allows accuracy in computations and shows respect for results where precision is paramount.

right triangle

A right triangle is a triangle with one angle measuring 90 degrees. It comprises three sides: the hypotenuse, opposite, and adjacent sides. Understanding this triangle is key to working with trigonometric functions.

• The hypotenuse is always the longest side and is opposite the right angle.
• The other sides are perpendicular to each other and form the angle of interest in the coordinate plane.

In this exercise, the point

• \((1.1, 6.0)\)

acts as the end of the perpendicular lines from the origin in the coordinate plane, creating a right triangle. Using these coordinate points, we calculate the hypotenuse as part of the trigonometric function calculations.

angle in standard position

An angle is in "standard position" when its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. This setup is foundational in trigonometry, as it provides a universal way to describe and compute angles and their corresponding trigonometric functions:

• The angle is measured from the positive x-axis to the terminal side, whether counterclockwise or clockwise.
• A positive measure represents counterclockwise rotation, while negative measures indicate clockwise rotation.

In this exercise, the terminal side passes through the point

• \((1.1, 6.0)\)

from the standard position. This positioning lets us use trigonometric identities linked to cos(θ), sin(θ), and tan(θ) in relation to the hypotenuse, adjacent, and opposite sides.