Problem 6 - Algebra and Trigonometry

Question

1-8 Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) from the given information. \(\sec x=2, \quad x\) in quadrant \(\mathrm{IV}\)

Step-by-Step Solution

Answer

\( \sin 2x = -\frac{\sqrt{3}}{2} \), \( \cos 2x = -\frac{1}{2} \), \( \tan 2x = \sqrt{3} \).

Step 1: Identify Given Information and Needed Formulas

We know that \( \sec x = 2 \), which means \( \cos x = \frac{1}{2} \) because \( \sec x = \frac{1}{\cos x} \). Since \( x \) is in quadrant IV, \( \cos x \) is positive and \( \sin x \) is negative. To find the double angle identities, we'll need the formulas: \[ \sin 2x = 2 \sin x \cos x \]\[ \cos 2x = \cos^2 x - \sin^2 x \]\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]

Step 2: Determine \( \sin x \) Using Trigonometric Identity

Use the identity \( \sin^2 x + \cos^2 x = 1 \) to find \( \sin x \): \[ \sin^2 x + \left( \frac{1}{2} \right)^2 = 1 \] \[ \sin^2 x + \frac{1}{4} = 1 \] \[ \sin^2 x = \frac{3}{4} \] Since \( x \) is in quadrant IV, \( \sin x \) is negative, thus:\[ \sin x = -\sqrt{\frac{3}{4}} = -\frac{\sqrt{3}}{2} \]

Step 3: Calculate \( \sin 2x \)

Now substitute \( \sin x = -\frac{\sqrt{3}}{2} \) and \( \cos x = \frac{1}{2} \) into the double angle formula for sine:\[ \sin 2x = 2 \sin x \cos x = 2 \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{1}{2}\right) = -\frac{\sqrt{3}}{2} \]

Step 4: Calculate \( \cos 2x \)

Use the double angle formula for cosine:\[ \cos 2x = \cos^2 x - \sin^2 x = \left(\frac{1}{2}\right)^2 - \left(-\frac{\sqrt{3}}{2}\right)^2 \] \[ = \frac{1}{4} - \frac{3}{4} = -\frac{1}{2} \]

Step 5: Calculate \( \tan 2x \)

Find \( \tan x \) first:\[ \tan x = \frac{\sin x}{\cos x} = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3} \] Then use the double angle formula for tangent:\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} = \frac{2(-\sqrt{3})}{1 - (-\sqrt{3})^2} = \frac{-2\sqrt{3}}{1 - 3} = \frac{-2\sqrt{3}}{-2} = \sqrt{3} \]

Key Concepts

Double Angle Formulas

Double angle formulas are important tools in trigonometry that allow you to express trigonometric functions of double angles, such as \(2x\), in terms of trigonometric functions of single angles \(x\). These formulas are particularly useful for simplifying trigonometric expressions and solving equations. The three primary double angle formulas are for sine, cosine, and tangent:

• \(\sin 2x = 2 \sin x \cos x\)
• \(\cos 2x = \cos^2 x - \sin^2 x\)
• \(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\)

Each formula transforms a problem involving an angle doubled, like \(2x\), into one involving simpler expressions tied to just \(x\). For example, to find \(\sin 2x\), you need both \(\sin x\) and \(\cos x\), which can initially seem complicated but eventually breaks down into elementary calculations once you know these values. Understanding and mastering these formulas allow you to tackle various types of trigonometric equations with ease.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are pivotal in the study of periodic phenomena such as waves and oscillations. Each of these functions has a reciprocal:

• Cosecant (\(\csc\)), the reciprocal of sine, calculated as \(\csc x = \frac{1}{\sin x}\).
• Secant (\(\sec\)), the reciprocal of cosine, seen as \(\sec x = \frac{1}{\cos x}\).
• Cotangent (\(\cot\)), the reciprocal of tangent, calculated using \(\cot x = \frac{1}{\tan x}\).

In solving any trigonometric problem, like the one above, identifying given information such as \(\sec x\) or \(\cos x\) can guide you towards finding other needed functions like \(\sin x\) or \(\tan x\) using fundamental identities and formulas. The interplay between these functions forms the backbone of solving complex trigonometric identities and equations.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four distinct quadrants, with each quadrant affecting the signs of the trigonometric functions differently. Quadrant I includes positive x and y values, Quadrant II has negative x and positive y, Quadrant III includes negative x and y values, and Quadrant IV has positive x and negative y values.

• In Quadrant I, all trigonometric functions are positive.
• In Quadrant II, sine is positive, while cosine and tangent are negative.
• In Quadrant III, tangent is positive, while sine and cosine are negative.
• In Quadrant IV, cosine is positive, whereas sine and tangent are negative.

It's crucial to know these sign conventions when solving trigonometric problems. For instance, in the given problem, knowing that the angle is in Quadrant IV helped determine that \(\cos x\) is positive and \(\sin x\) is negative. This distinction is vital for correctly applying trigonometric identities and calculating functions like \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) without error.