Problem 3 - Algebra and Trigonometry

Question

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\cos x=\frac{4}{5}, \quad \csc x<0\)

Step-by-Step Solution

Answer

\( \sin 2x = -\frac{24}{25}, \cos 2x = \frac{7}{25}, \tan 2x = -\frac{24}{7} \).

Step 1: Determine sin x using cosine

Given that \( \cos x = \frac{4}{5} \), we need to find \( \sin x \). Since \( \cos^2 x + \sin^2 x = 1 \), we have: \[ \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]\[ \sin x = \pm \frac{3}{5} \]. Since \( \csc x < 0 \) and \( \csc x = \frac{1}{\sin x} \), \( \sin x < 0 \). Thus, \( \sin x = -\frac{3}{5} \).

Step 2: Use sin and cos to find sin 2x

The formula for \( \sin 2x \) is \( 2 \sin x \cos x \). Substituting the values: \[ \sin 2x = 2 \left(-\frac{3}{5}\right)\left(\frac{4}{5}\right) = -\frac{24}{25} \].

Step 3: Use sin and cos to find cos 2x

The formula for \( \cos 2x \) is \( \cos^2 x - \sin^2 x \). Using the calculated values: \[ \cos 2x = \left(\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \].

Step 4: Use sin and cos to find tan 2x

The formula for \( \tan 2x \) is \( \frac{\sin 2x}{\cos 2x} \). Using the calculated \( \sin 2x \) and \( \cos 2x \): \[ \tan 2x = \frac{-\frac{24}{25}}{\frac{7}{25}} = -\frac{24}{7} \].

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the angles and sides of triangles. They enable us to simplify complex mathematical problems by expressing one trigonometric function in terms of others. A key identity used in many calculations is the Pythagorean identity:

• \( \cos^2 x + \sin^2 x = 1 \)

This identity helps us find unknown values like \( \sin x \) when \( \cos x \) is known. For example, when \( \cos x = \frac{4}{5} \), we can determine \( \sin x \) using the Pythagorean identity:

• \( \sin^2 x = 1 - \cos^2 x \)
• \( \sin x = \pm \frac{3}{5} \)

This process illustrates how trigonometric identities assist in solving equations and understanding trigonometric relationships effectively.

Sine Double Angle

The sine double angle formula is a powerful trigonometric identity that expresses \( \sin 2x \) in terms of \( \sin x \) and \( \cos x \). The formula is given by:

• \( \sin 2x = 2 \sin x \cos x \)

This formula is particularly useful when calculating the sine of a double angle without having to double the angle in the actual measurement. For instance, if we know \( \sin x = -\frac{3}{5} \) and \( \cos x = \frac{4}{5} \), it's straightforward to compute:

• \( \sin 2x = 2(-\frac{3}{5})(\frac{4}{5}) = -\frac{24}{25} \)

Breaking the formula down, we see that multiplying \( 2 \sin x \) by \( \cos x \) provides the sine of the double angle efficiently, demonstrating its utility in simplifying trigonometric expressions.

Cosine Double Angle

The cosine double angle formula provides a way to calculate \( \cos 2x \) using \( \sin x \) and \( \cos x \). The formula is expressed as:

• \( \cos 2x = \cos^2 x - \sin^2 x \)

Alternatively, it can be written using just one trigonometric function, such as:

• \( \cos 2x = 2\cos^2 x - 1 \)
• \( \cos 2x = 1 - 2\sin^2 x \)

These variations allow flexibility depending on the known values. For example, if \( \cos x = \frac{4}{5} \) and \( \sin x = -\frac{3}{5} \), using the first form we find:

• \( \cos 2x = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} \)

This showcases the cosine double angle formula's ability to derive angles easily from known trigonometric values.

Tangent Double Angle

The tangent double angle formula helps in determining \( \tan 2x \) from known sine and cosine values. The formula is as follows:

• \( \tan 2x = \frac{\sin 2x}{\cos 2x} \)

With values of \( \sin 2x \) and \( \cos 2x \) previously calculated, it simplifies finding \( \tan 2x \). Given \( \sin 2x = -\frac{24}{25} \) and \( \cos 2x = \frac{7}{25} \), we can compute:

• \( \tan 2x = \frac{-\frac{24}{25}}{\frac{7}{25}} = -\frac{24}{7} \)

This formula is incredibly useful as it consolidates the trigonometric process, streamlining computations that involve doubling the angle. Understanding and applying tangent double angle ensures accuracy and efficiency in solving trigonometric problems.