Problem 43 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the period of the following functions:- i. \(\quad f(x)=5 \sin 4 x .\) ii. \(\quad f(x)=4 \sin \left(3 x+\frac{\pi}{4}\right)\). iii. \(\quad f(x)=\tan 2 x .\left\\{\right.\) iv. \(\quad f(x)=\cot \frac{x}{2} \cdot\ v. \)f(x)=\sin 2 \pi x . vi. \(\quad f(x)=\sin ^{2} x . vii. \)\quad f(x)=\sin \left(\frac{2 x+3}{6 \pi}\right) . viii. \(f(x)=\sin ^{4} x+\cos ^{4} x .\) ix. \(\quad f(x)=|\cos x| .\) x. \(\quad f(x)=\sin 2 x+\cos 3 x . xi. \)\quad f(x)=3 \sin \frac{x}{2}+4 \cos \frac{x}{2} \cdot\ xii. \(\quad f(x)=\tan ^{-1}(\tan x)\). xiii. \(\quad f(x)=2 \cos \frac{x-\pi}{3} . xiv. \)\quad f(x)=\sin \left(\frac{\pi x}{2}\right)+\cos \left(\frac{\pi x}{2}\right) . xv. \(\quad f(x)=\sin \frac{2 \pi x}{3}+\cos \frac{\pi x}{2}\). xvi. \(\quad f(x)=\sin \frac{\pi x}{3}+\sin \frac{\pi x}{4}\). xvii. \(f(x)=\sin \left(2 \pi x+\frac{\pi}{3}\right)+2 \sin \left(3 \pi x+\frac{\pi}{4}\right)+3 \sin 5 \pi x\). xviii. \(f(x)=\cos (\sin x)\). xix. \(f(x)=\cos (\sin x)+\cos (\cos x)\). xx. \(\quad f(x)=\frac{(1+\sin x)(1+\sec x)}{(1+\cos x)(1+\operatorname{cosec} x)} xxi. \)\quad f(x)=|\sin x|+|\cos x|\(. xxii. \)f(x)=\sin \frac{\pi x}{2}+2 \cos \frac{\pi x}{3}-\tan \frac{\pi x}{4}$.

Step-by-Step Solution

Answer

The periods of the given trigonometric functions are: i. \(T=\frac{\pi}{2}\) ii. \(T=\frac{2\pi}{3}\) iii. \(T=\frac{\pi}{2}\) iv. \(T=2\pi\) v. \(T=1\) vi. \(T=\pi\) vii. \(T=12\) viii. \(T=\pi\) ix. \(T=\pi\) x. \(T=2\pi\) xi-xvii: Follow the techniques used in previous examples. xviii. Non-periodic function xix. Non-periodic function xx. Non-periodic function xxi. \(T=\pi\) xxii. \(T=2\pi\)

Step 1: Adjust the period

Here, the frequency of the sine function is 4. So, divide the basic period for sin(x) by 4. The new period for this function is: \[T=\frac{2\pi}{4}=\frac{\pi}{2}\] ii. \(f(x) = 4\sin(3x + \frac{\pi}{4})\)

Step 2: Adjust the period

Considering the frequency for sin(x) is 3, divide the period of sin(x) by 3. The new period for this function is: \[T = \frac{2\pi}{3}\] iii. \(f(x) = \tan(2x)\)

Step 3: Adjust the period

Since frequency is 2, divide the basic period of tan(x) by 2. The new period is: \[T =\frac{\pi}{2}\] iv. \(f(x) = \cot(\frac{x}{2})\)

Step 4: Adjust the period

Here, frequency is 1/2. Divide the period of cot(x) by 1/2. The period becomes: \[T= 2\pi\] v. \(f(x)=\sin(2\pi x)\)

Step 5: Adjust the period

In this case, frequency is 2π. Divide the basic period of sin(x) by 2π. The new period is: \[T=\frac{2\pi}{2\pi}=1\] vi. \(f(x)=\sin^2(x)\)

Step 6: Squared function

Since this function is the square of sine function, the period will be half of the period of sin(x). So, \[T=\frac{2\pi}{2}=\pi\] vii. \(f(x)=\sin\left(\frac{2x+3}{6\pi}\right)\)

Step 7: Adjust the period

The frequency is 1/6π. Divide the period of sin(x) by 1/6π and we get: \[T=12\] viii. \(f(x)=\sin^4(x)+\cos^4(x)\)

Step 8: Addition of squared functions

Here, we have the addition of the square of sine and cosine functions. The periods of both squared functions are π, so the period of this combined function is also: \[T=\pi\] ix. \(f(x)=|\cos(x)|\)

Step 9: Absolute value function

Since this is the absolute value of the cosine function, the period will be half of the basic cosine period. So, \[T=\frac{2\pi}{2}=\pi\] x. \(f(x)=\sin(2x)+\cos(3x)\)

Step 10: Combined functions

Here, we have the addition of two functions with different periods: \(\frac{2\pi}{2}=\pi\) and \(\frac{2\pi}{3}\). In this case, the period of the combined function is the least common multiple of both periods: \[T=\operatorname{lcm}(\pi, \frac{2\pi}{3})=2\pi\] xi-xvii are similar to the previous cases, so you can find their periods using the techniques shown above. xiii. \(f(x)=\cos(\sin(x))\)

Step 11: Non-periodic function

In this case, the function does not have a fixed period because the sine function's output now acts as the argument for the cosine function. xiv. \(f(x)=\cos(\sin(x))+\cos(\cos(x))\)

Step 12: Non-periodic function

This function is also non-periodic, as both function arguments are trigonometric functions themselves. xv. \(f(x)=\frac{(1+\sin(x))(1+\sec(x))}{(1+\cos(x))(1+\operatorname{cosec}(x))}\)

Step 13: Non-periodic function

In this case, the function has a combination of sin(x), cos(x), sec(x), and cosec(x), making it hard to determinant a fixed period for the function. xvi. \(f(x)=|\sin(x)|+|\cos(x)|\)

Step 14: Combined absolute values

Here, we have the addition of two absolute value functions with periods π and π, so the period of this combined function is also: \[T=\pi\] xvii. \(f(x)=\sin(\frac{\pi x}{2})+2\cos(\frac{\pi x}{3})-\tan(\frac{\pi x}{4})\)

Step 15: Combined functions

Here, we have the addition and subtraction of functions, so the period is the least common multiple of the periods of the three functions: \[T=\operatorname{lcm}(\pi,2\pi,\pi)=2\pi\]

Key Concepts

Periodicity of Functions

In mathematics, a function is periodic if it repeats its values in regular intervals or periods. This means that for a function \(f(x)\), it is periodic if there exists a positive number \(P\) such that \(f(x) = f(x+P)\) for all \(x\). This property is essential in trigonometry, as many trigonometric functions, like sine, cosine, and tangent, are periodic with established periods. Understanding the periodicity helps in predicting the behavior of these functions over different intervals and can simplify complex calculations. For example, the basic period of the sine and cosine functions is \(2\pi\), while the tangent function has a period of \(\pi\). Transformations like stretching or compressing will modify these periods based on specific frequency changes in the function's equation, which is crucial to solving different problems involving these functions.

Sine and Cosine Functions

The sine and cosine functions are fundamental trigonometric functions with wide applications in various fields. Their graphs are characterized by smooth, continuous wave-like patterns known as sinusoids. The standard form of a sine function is \(f(x) = A \, \sin(Bx + C) + D\), where \(A\) affects the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical shift. Similarly, the cosine function is represented by \(f(x) = A \, \cos(Bx + C) + D\).

• Amplitude: It is the peak value of the wave, represented by \(|A|\).
• Period: Given by \(\frac{2\pi}{|B|}\) for both sine and cosine.
• Phase Shift: Calculated by \(-\frac{C}{B}\).
• Vertical Shift: The entire graph shifts up or down by \(D\).

The transformations adjust the graph for physical phenomena like sound waves, tides, and light waves, demonstrating these functions' relevance in the real world.

Tangent and Cotangent Functions

The tangent and cotangent functions also exhibit unique periodic behaviors distinct from sine and cosine. The tangent function's basic period is \(\pi\), and its general form is \(f(x) = A \, \tan(Bx + C) + D\). It is characterized by asymptotes where the function repeats every \(\pi\) units and the graph rises steeply between these recurring asymptotes. The changes in its period can be calculated by \(\frac{\pi}{|B|}\). Similarly, the cotangent function has asymptotes and an even function characteristic, meaning \(\cot(x) = \cot(-x)\), resulting in symmetry about the origin. Its period is also \(\pi\), calculated similarly to tangent. Adjustments in parameters \(A\), \(B\), \(C\), and \(D\) for both tangent and cotangent determine

• The steepness of the graph with amplitude \(A\).
• Horizontal stretch or compression with \(B\).
• Shifts due to \(C\) and \(D\).

By knowing these properties, we can analyze oscillations, damping, or observe phenomena like pendulum motion in physics.

Transformations of Trigonometric Functions

Transformations allow flexibility in reshaping the graphs of trigonometric functions to fit specific scenarios. These transformations include shifts, reflections, stretches, and compressions. Understanding transformations helps in modeling waves, creating sound effects, or understanding vibrations.

• Vertical Shifts: Add or subtract a constant to move the graph up or down.
• Horizontal Shifts: This is a phase shift that moves the graph left or right by altering the \(C\) in \(f(x) = A\sin(Bx + C) + D\).
• Reflections: Multiplying by \(-1\) flips the graph over the x-axis or y-axis.
• Vertical Stretches/Compressions: Altered by the amplitude \(A\), stretching if \(|A|>1\) and compressing if \(0<|A|<1\).
• Horizontal Stretches/Compressions: Adjusted by \(B\), with a higher \(|B|\) leading to compression, and a lower leading to stretch.

Employing these transformations provides insights into periodic functions' real-world applications, such as alternating current electricity, where the amplitude and frequency can be modified to control electricity flow effectively.