Problem 33 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos 2 x,(b) \sin 2 x\), and \((c) \tan 2 x\). $$ \sec x=-\frac{13}{5}, \quad \pi / 2

Step-by-Step Solution

Answer

\(\cos 2x = -\frac{119}{169}, \sin 2x = -\frac{120}{169}, \tan 2x = \frac{120}{119}\)

Step 1: Identify the Quadrant

The problem states that \( \frac{\pi}{2} < x < \pi \), which places \( x \) in the second quadrant. In this quadrant, \( \cos x < 0 \), \( \sin x > 0 \), and \( \tan x < 0 \).

Step 2: Find \( \cos x \)

We know that \( \sec x = -\frac{13}{5} \). The secant function is the reciprocal of the cosine function, so \( \cos x = -\frac{5}{13} \).

Step 3: Use Pythagorean Identity to Find \( \sin x \)

Using the identity \( \cos^2 x + \sin^2 x = 1 \), substitute \( \cos x = -\frac{5}{13} \): \[ \left( -\frac{5}{13} \right)^2 + \sin^2 x = 1 \]\[ \frac{25}{169} + \sin^2 x = 1 \]\[ \sin^2 x = 1 - \frac{25}{169} = \frac{144}{169} \]\( \sin x = \frac{12}{13} \) (since sine is positive in the second quadrant).

Step 4: Find \( \cos 2x \) Using Double Angle Formula

The formula for \( \cos 2x \) is \( \cos 2x = \cos^2 x - \sin^2 x \). Substitute the values we found:\[ \cos 2x = \left( -\frac{5}{13} \right)^2 - \left( \frac{12}{13} \right)^2 \]\[ \cos 2x = \frac{25}{169} - \frac{144}{169} = -\frac{119}{169} \]

Step 5: Find \( \sin 2x \) Using Double Angle Formula

The formula for \( \sin 2x \) is \( \sin 2x = 2 \sin x \cos x \). Substitute the values:\[ \sin 2x = 2 \left( \frac{12}{13} \right) \left( -\frac{5}{13} \right) \]\[ \sin 2x = 2 \cdot \frac{12}{13} \cdot -\frac{5}{13} = -\frac{120}{169} \]

Step 6: Find \( \tan 2x \) Using Double Angle Formula

The formula for \( \tan 2x \) is \( \tan 2x = \frac{\sin 2x}{\cos 2x} \). Substitute the values:\[ \tan 2x = \frac{-\frac{120}{169}}{-\frac{119}{169}} = \frac{120}{119} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true. These identities are essential in solving and simplifying trigonometry problems. They allow you to express trigonometric functions in different ways, making calculations more manageable. One important identity is the Pythagorean identity: \[\cos^2 x + \sin^2 x = 1\]This identity helps relate the sine and cosine functions of an angle and is useful when you have one function and need to find the other.Another set of identities are the double angle formulas. They let you find the trigonometric functions of double angles. For instance, the double angle formulas used in the original problem are:

• \(\cos 2x = \cos^2 x - \sin^2 x\)
• \(\sin 2x = 2 \sin x \cos x\)
• \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\), though in the solution, we used it as \(\tan 2x = \frac{\sin 2x}{\cos 2x}\).

These formulas are handy in bridging the gap between single and double angles, aiding in solving complex problems.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, serve as the building blocks in trigonometry. They relate the angles of a triangle to the ratios of its sides and are defined on the unit circle.For example, in the unit circle:

• \(\cos \theta\) gives the x-coordinate of the point on the unit circle corresponding to the angle \(\theta\).
• \(\sin \theta\) provides the y-coordinate.
• \(\tan \theta\) is the ratio \(\frac{\sin \theta}{\cos \theta}\).

Additionally, the secant (\(\sec x\)) is the reciprocal of the cosine function, meaning \(\sec x = \frac{1}{\cos x}\). Understanding the relationships among these functions is crucial when moving between different trigonometric forms or when given constraints like \(\sec x = -\frac{13}{5}\). This was used in the original problem's solution to find \(\cos x\) as \(-\frac{5}{13}\).

Second Quadrant Angles

Angles in the second quadrant of the unit circle range from \(\frac{\pi}{2}\) to \(\pi\). In this quadrant, certain properties apply:

• \(\sin x > 0\): The sine of an angle is positive because the y-coordinate of points on the unit circle is positive in this region.
• \(\cos x < 0\): The cosine of an angle is negative since the x-coordinates are negative.
• \(\tan x < 0\): The tangent, being the ratio of sine to cosine, is negative because you're dividing a positive by a negative.

Understanding these signs is critical for correctly applying the trigonometric identities and for knowing the properties of the trigonometric functions in this region. For example, recognizing that \(\cos x\) is negative helped us identify that \(\sec x = -\frac{13}{5}\) translates to \(\cos x = -\frac{5}{13}\). These properties also ensure correct usage of the Pythagorean identity to find \(\sin x = \frac{12}{13}\).