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What is the Transformation of a cos x + b sin x into R sin (x + α)?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The transformation of a cos x + b sin x into R sin (x + α) is a method to combine two trigonometric terms into a single, simpler sine function. This helps us easily find the maximum or minimum value of the expression and understand its wave-like behaviour. Here, 'R' represents the amplitude (maximum value) and 'α' (alpha) is the phase shift.
Simple Example
Quick Example
Imagine you're trying to figure out the total 'strength' of two different mobile network signals, one varying with cos x and the other with sin x. Instead of tracking two separate signals, this transformation helps you combine them into one single signal, R sin (x + α), making it much easier to see the overall signal strength and when it's strongest. It's like combining two small 'dabba' radios into one powerful music system!
Worked Example
Step-by-Step
Let's transform 3 cos x + 4 sin x into the form R sin (x + α).
Step 1: Compare 3 cos x + 4 sin x with R sin (x + α) = R (sin x cos α + cos x sin α).
--- Step 2: Rearrange R sin (x + α) to R cos α sin x + R sin α cos x. Now, by comparing coefficients, we get: R cos α = 4 (coefficient of sin x) and R sin α = 3 (coefficient of cos x).
--- Step 3: Square both equations and add them: (R cos α)^2 + (R sin α)^2 = 4^2 + 3^2. This gives R^2 (cos^2 α + sin^2 α) = 16 + 9.
--- Step 4: Since cos^2 α + sin^2 α = 1, we have R^2 * 1 = 25. So, R^2 = 25, which means R = 5 (amplitude is always positive).
--- Step 5: Divide the two equations: (R sin α) / (R cos α) = 3 / 4. This simplifies to tan α = 3 / 4.
--- Step 6: Find α. Since sin α and cos α (from R sin α = 3 and R cos α = 4) are both positive, α is in the first quadrant. α = arctan (3/4) ≈ 36.87 degrees.
--- Step 7: Substitute R and α back into the form. So, 3 cos x + 4 sin x = 5 sin (x + 36.87 degrees).
Answer: 3 cos x + 4 sin x transforms into 5 sin (x + 36.87 degrees).
Why It Matters
This transformation is super useful in understanding waves and oscillations, which are everywhere from sound to light to earthquake tremors. Engineers use it to design stable bridges and buildings, and physicists use it to analyze signal processing in areas like mobile communication or even space technology. Understanding this helps you think like a scientist or engineer!
Common Mistakes
MISTAKE: Confusing the coefficients. Students often mistakenly set R cos α equal to the coefficient of cos x and R sin α to the coefficient of sin x. | CORRECTION: Remember the expansion R sin (x + α) = R sin x cos α + R cos x sin α. So, R cos α is the coefficient of sin x, and R sin α is the coefficient of cos x. (Or if using R cos (x - α), then it's R cos x cos α + R sin x sin α, so R cos α is with cos x and R sin α is with sin x). Be consistent!
MISTAKE: Calculating alpha (α) incorrectly, especially its quadrant. Students might just use the calculator value for arctan without checking the signs of R sin α and R cos α. | CORRECTION: After finding R sin α and R cos α, check their signs. If both are positive, α is in Quadrant I. If R sin α is positive and R cos α is negative, α is in Quadrant II, and so on. This ensures you get the correct phase shift.
MISTAKE: Forgetting that R must be positive. Students sometimes take the negative square root when finding R. | CORRECTION: R represents the amplitude of the wave, which is a magnitude and must always be a positive value. So, R = sqrt(a^2 + b^2) will always be positive.
Practice Questions
Try It Yourself
QUESTION: Transform 5 cos x + 12 sin x into the form R sin (x + α). | ANSWER: 13 sin (x + 22.62 degrees)
QUESTION: Transform sqrt(3) sin x + cos x into the form R sin (x + α). | ANSWER: 2 sin (x + 30 degrees)
QUESTION: A sound wave's displacement is given by y = 2 sin x - 2 cos x. Transform this into R sin (x - α) and find its maximum displacement. | ANSWER: 2*sqrt(2) sin (x - 45 degrees); Maximum displacement = 2*sqrt(2)
MCQ
Quick Quiz
What is the value of R when transforming 6 cos x + 8 sin x into R sin (x + α)?
6
8
10
14
The Correct Answer Is:
C
R is calculated as sqrt(a^2 + b^2). Here, a=6 and b=8. So, R = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10.
Real World Connection
In the Real World
Imagine the electrical power grid in your city. The voltage and current often vary in a wave-like pattern. Electrical engineers use this transformation to combine different AC (Alternating Current) signals from various power sources into a single, manageable signal. This helps them balance the load, prevent power outages, and ensure your home always gets a steady supply of electricity, just like how ISRO scientists combine signals from different satellite antennas!
Key Vocabulary
Key Terms
Amplitude: The maximum displacement or distance moved by a point on a vibrating body or wave, measured from its equilibrium position. | Phase Shift: A horizontal shift of the graph of a trigonometric function. | Trigonometric Identity: An equation involving trigonometric functions that is true for all values of the variables for which the functions are defined. | Coefficient: A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
What's Next
What to Learn Next
Great job mastering this! Next, you can explore how to transform a cos x + b sin x into R cos (x - α) or R cos (x + α). This will further strengthen your understanding of wave forms and prepare you for more advanced topics in physics and engineering, making you ready to tackle complex problems!
