Problem 5 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(|\sin 2 x+\cos 2 x|=|\sin 2 x|+|\cos 2 x|\)

Step-by-Step Solution

Answer

The validity of the given equation \(|\sin 2 x+\cos 2 x|=|\sin 2 x|+|\cos 2 x|\) depends on the value of \(x\). The equation is true when the signs of \(\sin 2x\) and \(\cos 2x\) are the same, and false otherwise.

Step 1: Identify the ranges

Identify the ranges of \(x\) where \(\sin 2x\) and \(\cos 2x\) are positive and negative. Remember that \(\sin x\) is positive in the 1st and 2nd quadrants, and \(\cos x\) is positive in the 1st and 4th quadrants. Given that we are dealing with \(2x\), this divided the intervals: \([0, \frac{π}{2})\), \([\frac{π}{2}, π)\), \([π, \frac{3π}{2})\), \([\frac{3π}{2}, 2π)\) and so on.

Step 2: Substitute and Verify

Substitute any value of \(x\) within each identified interval into the expression \(|\sin 2x+\cos 2x|=|\sin 2x|+|\cos 2x|\) and verify whether the equation is true or not.

Step 3: Define domains and examine the equality

Based on the results from step 2, define the domains where \(|\sin 2 x+\cos 2 x|=|\sin 2 x|+|\cos 2 x|\) is true and where it is false. You find the valid regions by examining where the trigonometric functions are consistent, both in magnitude and sign.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is denoted by \( \sin(x) \) and represents the y-coordinate of a point on the unit circle. The sine function has a range from -1 to 1, meaning it can take any value between these two. The function is periodic with a period of \( 2\pi \), which means it repeats its values every \( 2\pi \) units.

• Positive and Negative: The sine function is positive in the first and second quadrants of the unit circle. In these quadrants, the y-coordinate is above the x-axis. It is negative in the third and fourth quadrants, where the y-coordinate is below the x-axis.
• Zero Points: The sine function is zero at integer multiples of \( \pi \), such as \( 0, \pi, 2\pi \), etc.

Understanding the behavior of the sine function is crucial when solving problems involving trigonometric identities, especially when working with equations like the one in the exercise—where the absolute values of the sine function and its complements are considered.

Cosine Function

The cosine function is another principal trigonometric function, denoted by \( \cos(x) \). It represents the x-coordinate of a point on the unit circle. Just like the sine function, it ranges from -1 to 1 and is periodic with a period of \( 2\pi \).

• Positive and Negative: The cosine function is positive in the first and fourth quadrants of the unit circle. These are the sections where the x-coordinate is positive. It turns negative when the angle is in the second and third quadrants, where the x-coordinate is negative.
• Zero Points: The cosine function is zero at angles like \( \frac{\pi}{2}, \frac{3\pi}{2} \), where the point lies on the y-axis.

When working with problems that incorporate cosine, it helps to know how changes in the angle affect the output of the cosine function. In the provided trigonometric equation, recognizing where the cosine function is positive or negative can help determine the validity of the equality across different intervals.

Quadrants in Trigonometry

Quadrants in trigonometry divide the coordinate plane into four areas based on the signs of the sine, cosine, and tangent functions. Each quadrant determines whether these trigonometric functions are positive or negative based on the angle's position.

• First Quadrant (0 to \( \frac{\pi}{2} \)): Both sine and cosine are positive. This is the "all positive" quadrant.
• Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)): Sine is positive, but cosine is negative. This is because the angle is above the x-axis and to the left.
• Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)): Both sine and cosine are negative. The angles here put the points in the left-bottom.
• Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)): Sine is negative, but cosine is positive. Angles reside above the x-axis, but to the right of the y-axis.

In analyzing equations that involve sine and cosine, like the one provided in the exercise, it is essential to consider which quadrant the angle falls in. This aids in determining the signs of the respective trigonometric functions and thus the truthfulness of expressions.