Problem 2 - Contemporary Precalculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\sin x=-\frac{4}{5} \quad\left(\pi

Step-by-Step Solution

Answer

Question: Find the values of \(\sin 2x, \cos 2x,\) and \(\tan 2x\) given that \(\sin x = -\frac{4}{5}\) and \(\pi < x < \frac{3\pi}{2}\). Answer: \(\sin 2x = \frac{24}{25}, \cos 2x = -\frac{7}{25}, \tan 2x = -\frac{24}{7}\)

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

Since the given condition is \(\sin x = -\frac{4}{5}\), we can find the value of \(\cos x\) by using the Pythagorean identity: \(\sin^2 x +\cos^2 x =1\). Plugging in \(\sin x\) value, we find: $$\cos^2 x= 1 - (-\frac{4}{5})^2$$ $$\cos^2 x= 1 - \frac{16}{25}$$ $$\cos^2 x= \frac{9}{25}$$ As \(\cos x\) is negative in the given interval (\(\pi < x < \frac{3\pi}{2}\)), we get \(\cos x = -\frac{3}{5}\).

Step 2: Calculate \(\sin 2x\) using Double-Angle identity

We'll use the double-angle formula for the sine function: \(\sin 2x = 2 \sin x \cos x\). Now, plug the values of \(\sin x\) and \(\cos x\) into the equation: $$\sin 2x = 2 \cdot -\frac{4}{5} \cdot -\frac{3}{5}$$ $$\sin 2x = \frac{24}{25}$$

Step 3: Calculate \(\cos 2x\) using Double-Angle identity

We'll use the double-angle formula for the cosine function: \(\cos 2x = \cos^2 x - \sin^2 x\). Now, plug the values of \(\sin x\) and \(\cos x\) into the equation: $$\cos 2x = (-\frac{3}{5})^2 - (-\frac{4}{5})^2$$ $$\cos 2x = \frac{9}{25} - \frac{16}{25}$$ $$\cos 2x = -\frac{7}{25}$$

Step 4: Calculate \(\tan 2x\) using Double-Angle identity

We'll use the double-angle formula for the tangent function: \(\tan 2x = \frac{\sin 2x}{\cos 2x}\). Now, plug the values of \(\sin 2x\) and \(\cos 2x\) into the equation: $$\tan 2x = \frac{\frac{24}{25}}{-\frac{7}{25}}$$ $$\tan 2x = -\frac{24}{7}$$ #Conclusion# Under the given conditions, we have found out that \(\sin 2x = \frac{24}{25}\), \(\cos 2x = -\frac{7}{25}\), and \(\tan 2x = -\frac{24}{7}\).

Key Concepts

Double-Angle Formulas

Double-angle formulas are crucial tools in trigonometry. They help us express trigonometric functions of doubled angles, such as \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\), using the functions of \(x\). These formulas are derived from sum identities and simplify many problems involving angles.

• For sine, the double-angle formula is \(\sin 2x = 2 \sin x \cos x\). This gives us a way to find the sine of a double angle by multiplying the sine and cosine of the single angle and then doubling the result.


• The cosine formula is \(\cos 2x = \cos^2 x - \sin^2 x\). This illustrates that the cosine of a double angle is the difference of squares of the cosine and sine.


• For tangent, the formula is \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\), which helps in finding the tangent of a doubled angle by knowing the tangent of the angle.

By using these formulas, it becomes easier to solve trigonometric equations and understand angular relationships.

Pythagorean Identity

The Pythagorean identity is one of the fundamental building blocks in trigonometry. It comes from the Pythagorean theorem applied to a unit circle. The identity tells us that:

• \(\sin^2 x + \cos^2 x = 1\)

This equation is incredibly helpful when trying to find unknown trigonometric values.In this exercise, we used the identity to find \(\cos x\) when \(\sin x = -\frac{4}{5}\). Knowing \(\sin x\), we can rearrange the identity to find \(\cos x\) by:

• \(\cos^2 x = 1 - \sin^2 x\)

Substituting the value of \(\sin x\), we calculated \(\cos x\) and determined its sign based on the given interval for \(x\). This process is vital for understanding relationships within the unit circle and solving problems accurately.

Trigonometric Functions

Trigonometric functions like \(\sin x\), \(\cos x\), and \(\tan x\) describe angles in terms of ratios. These functions are essential for understanding wave patterns, sound, light, and cycles in nature.Here's a brief overview:

• Sine (\(\sin x\)): Represents the ratio of the opposite side to the hypotenuse in a right triangle.


• Cosine (\(\cos x\)): Represents the ratio of the adjacent side to the hypotenuse.


• Tangent (\(\tan x\)): Calculated as the ratio of \(\sin x\) to \(\cos x\), representing the slope of the angle in the unit circle.

These functions help model periodic phenomena and solve real-life problems. In the given problem, we explored their double-angle counterparts to solve for angles precisely, showing their utility and interconnectedness in trigonometric reasoning.