Problem 15 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

Period of \(f(x)=\sin \frac{\pi}{2} x+2 \cos \frac{\pi}{3} x-\tan \frac{\pi}{4} x\) is equal to (a) 4 (b) 8 (c) 12 (d) 16

Step-by-Step Solution

Answer

The period of the function is 12.

Step 1: Identify the Periods of Each Trigonometric Function

To find the period of a trigonometric function, use the formula for sine and cosine: \( T = \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). For tangent, use \( T = \frac{\pi}{b} \). Here, we get:\- \( \sin \frac{\pi}{2} x \) has \( b = \frac{\pi}{2} \), so its period \( T_1 = \frac{2\pi}{\frac{\pi}{2}} = 4 \).\- \( \cos \frac{\pi}{3} x \) has \( b = \frac{\pi}{3} \), so its period \( T_2 = \frac{2\pi}{\frac{\pi}{3}} = 6 \).\- \( -\tan \frac{\pi}{4} x \) has \( b = \frac{\pi}{4} \), so its period \( T_3 = \frac{\pi}{\frac{\pi}{4}} = 4 \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The period of the function is 12.

Key Concepts

The Sine Function

The sine function is one of the basic trigonometric functions used to model periodic phenomena.Its general form is given by \( f(x) = \sin(bx) \), where the parameter \( b \) determines the function's period. This period is the length of one complete cycle of the wave, the time it takes for the curve to start repeating itself. For the sine function, the period is calculated using the formula \( T = \frac{2\pi}{b} \).

In simpler terms, this means that the graph of the sine function will start to repeat every \( T \) units.This periodic behavior is why sine is very handy in modeling repeating phenomena, such as sound waves or circular motion.For instance, when dealing with \( f(x) = \sin \frac{\pi}{2} x \), we identify \( b = \frac{\pi}{2} \), yielding a period \( T_1 = 4 \).This indicates that every 4 units along the x-axis, the sine wave repeats its pattern.

• Always check the coefficient of \( x \) to find the period.
• Sine waves are symmetrical and have a consistent waving pattern useful in various applications.
• Understanding periods helps with predicting and graphing the function accurately.

The Cosine Function

Much like the sine function, the cosine function is crucial in trigonometry. Its general form is \( f(x) = \cos(bx) \), similarly relying on the coefficient \( b \) to determine its period.The formula for the period of the cosine function is the same as that for sine: \( T = \frac{2\pi}{b} \).Although cosine also portrays periodicity, its wave pattern starts at a peak point, or a maximum value.

For example, with \( \cos \frac{\pi}{3} x \), we use \( b = \frac{\pi}{3} \), which gives us a period of \( T_2 = 6 \).This tells us that the graph of this function completes a cycle every 6 units along the x-axis, meaning it returns to the starting point consistently over this distance.

• Cosine and sine functions are phase-shifted versions of each other (one starts at a peak and the other at a mid-point).
• Both functions have amplitudes and frequencies, but here we focus on period using \( b \).
• Mastering the period concept is vital for accurately plotting and interpreting these functions.

The Tangent Function

The tangent function is a bit different from sine and cosine regarding periodicity.Its general form \( f(x) = \tan(bx) \) has a different period calculation: \( T = \frac{\pi}{b} \).Instead of completing one cycle over \( 2\pi \), the tangent function does so over \( \pi \).

In our case with \( -\tan \frac{\pi}{4} x \), the coefficient \( b = \frac{\pi}{4} \) suggests a period of \( T_3 = 4 \).This means the graph of the tangent function finishes a full cycle every 4 units on the x-axis.The nature of the tangent function includes asymptotes within each period, where the function tends to infinity or negative infinity, a character not seen in the sine and cosine.

• Tangent's period calculation is different because it completes a cycle faster than sine and cosine functions.
• Unlike sine and cosine, tangent graphs do not rest at their maximum or minimum but continue beyond to infinity.
• Understanding tangent's longer cycle and asymptotic behavior is important for practical application in wave forms and oscillation principles.