Problem 45 - Contemporary Precalculus

Question

Assume sin \(x=.6\) and \(0

Step-by-Step Solution

Answer

Question: Given that \(\sin x = 0.6\) and \(0 < x < \frac{\pi}{2}\), find the value of \(\sin 2x\). Answer: \(\sin 2x = 0.96\)

Step 1: Find the value of \(\cos x\)

Since we know \(\sin x = 0.6\) and \(0 < x < \frac{\pi}{2}\), we can use the Pythagorean identity \(\sin^2 x + \cos^2x = 1\) to find the value of \(\cos x\). $$ \cos^2 x = 1 - \sin^2 x = 1 - 0.6^2 = 1 - 0.36 = 0.64 $$ Now take the positive square root of \(0.64\) (We take the positive square root because \(0 < x < \frac{\pi}{2}\) which means \(\cos x\) is positive in the first quadrant). $$ \cos x = \sqrt{0.64} = 0.8 $$

Step 2: Use the double angle formula to find \(\sin 2x\)

Now that we have the values of both \(\sin x\) and \(\cos x\), we can use the double angle formula for sine to find the value of \(\sin 2x\): $$ \sin 2x = 2 \sin x \cos x = 2 \cdot 0.6 \cdot 0.8 = 2 \cdot 0.48 = 0.96 $$ Therefore, the value of \(\sin 2x\) is \(0.96\).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine functions. The identity states:

• \( \sin^2 x + \cos^2 x = 1 \)

This equation is incredibly useful for finding one trigonometric function, given the value of another, especially when the angle is in standard position in the coordinate plane.
In the given exercise, knowing that \(\sin x = 0.6\), we apply the Pythagorean identity to find \(\cos x\).
Rearranging the identity gives us \(\cos^2 x = 1 - \sin^2 x\).
Substituting the known value of sine gives:

• \( \cos^2 x = 1 - 0.6^2 = 0.64 \)

Since \(x\) is in the first quadrant, both sine and cosine are positive.
Therefore, \(\cos x = \sqrt{0.64} = 0.8\).
Understanding and applying the Pythagorean identity provides a quick way to establish relationships between trigonometric functions.

Double Angle Formula

The double angle formulas are trigonometric identities that express trigonometric functions of double angles \(2x\) in terms of a single angle \(x\). For example, the double angle formula for sine is:

• \( \sin 2x = 2 \sin x \cos x \)

In the context of the exercise, this formula helps us calculate \(\sin 2x\) using the values of \(\sin x\) and \(\cos x\).
Given \(\sin x = 0.6\) and \(\cos x = 0.8\), the calculation is straightforward:

• \( \sin 2x = 2 \times 0.6 \times 0.8 \)
• \( \sin 2x = 2 \times 0.48 = 0.96 \)

This outcome demonstrates how double angle formulas offer a powerful tool for simplifying and solving problems involving trigonometric expressions with multiple angles.
They are widely used in trigonometry to model phenomena such as oscillations and waves.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle, and they play a pivotal role in the study of triangles. These functions, including sine, cosine, and tangent, are periodic and are fundamental in describing geometrical relationships.
The main functions are:

• Sine (\(\sin x\)) - represents the opposite side over the hypotenuse in a right triangle.
• Cosine (\(\cos x\)) - represents the adjacent side over the hypotenuse.
• Tangent (\(\tan x\)) - represents the ratio of the opposite side to the adjacent side.

In the context of the exercise, understanding sine and cosine is crucial since they are directly used to simplify expressions like \(\sin 2x\).
The functions exhibit specific properties in different quadrants on the unit circle, with both sine and cosine being positive in the first quadrant.
Knowledge of these trigonometric functions and their properties allows for the solving of complex equations and the modeling of real-world phenomena through mathematics.