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What is the Transformation of a cos x + b sin x into R cos (x - α)?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
This transformation helps us combine two trigonometric terms, 'a cos x' and 'b sin x', into a single, simpler form 'R cos (x - α)'. It's like converting two separate ingredients into one easy-to-handle mixture. This makes it much easier to find maximum/minimum values, solve equations, or graph the function.
Simple Example
Quick Example
Imagine you have two friends, one gives you 3 toffees and the other gives you 4 chocolates, but you want to know the total 'sweetness' in one combined unit. This transformation is similar: 'a' and 'b' are like the number of toffees/chocolates, 'cos x' and 'sin x' are their 'sweetness' types, and 'R cos (x - α)' is the total combined 'sweetness' in a single, comparable form.
Worked Example
Step-by-Step
Let's transform 3 cos x + 4 sin x into the form R cos (x - α).
Step 1: Compare 3 cos x + 4 sin x with R cos (x - α) = R (cos x cos α + sin x sin α).
---Step 2: By comparing, we get R cos α = 3 and R sin α = 4.
---Step 3: To find R, square both equations and add them: (R cos α)^2 + (R sin α)^2 = 3^2 + 4^2. This gives R^2 (cos^2 α + sin^2 α) = 9 + 16. Since cos^2 α + sin^2 α = 1, we have R^2 = 25. So, R = sqrt(25) = 5.
---Step 4: To find α, divide the second equation by the first: (R sin α) / (R cos α) = 4 / 3. This simplifies to tan α = 4/3. So, α = arctan(4/3). (You can use a calculator to find α approximately 53.13 degrees).
---Step 5: Substitute R and α back into the form. So, 3 cos x + 4 sin x = 5 cos (x - arctan(4/3)).
Answer: 3 cos x + 4 sin x transforms into 5 cos (x - arctan(4/3)).
Why It Matters
This transformation is super useful in fields like Physics to understand wave motion and sound, or in Engineering to analyze electrical circuits. Engineers and scientists use this to simplify complex signals, which helps in designing everything from mobile phone networks to medical imaging equipment.
Common Mistakes
MISTAKE: Forgetting to square 'a' and 'b' when finding R, or taking the square root of only one term. | CORRECTION: Remember R = sqrt(a^2 + b^2). You must square both 'a' and 'b' first, add them, and then take the square root of the sum.
MISTAKE: Incorrectly determining the quadrant of alpha (α) or using the wrong sign for 'a' or 'b'. | CORRECTION: Always consider the signs of 'a' (R cos α) and 'b' (R sin α) to correctly place α in its quadrant. For example, if 'a' is positive and 'b' is negative, α will be in the fourth quadrant.
MISTAKE: Confusing the formula for R cos (x - α) with R sin (x + α) or other forms. | CORRECTION: Stick to the standard expansion: R cos (x - α) = R (cos x cos α + sin x sin α) for comparison. If the question asks for a different form, expand that specific form carefully.
Practice Questions
Try It Yourself
QUESTION: Transform 5 cos x + 12 sin x into R cos (x - α). | ANSWER: 13 cos (x - arctan(12/5))
QUESTION: Transform 2 sin x - 2 cos x into R cos (x - α). Hint: Rearrange to match the a cos x + b sin x form first. | ANSWER: 2sqrt(2) cos (x - arctan(-1)) which is 2sqrt(2) cos (x + pi/4) or 2sqrt(2) cos (x + 45 degrees)
QUESTION: If 6 cos x + 8 sin x = R cos (x - α), find the maximum value of the expression and the value of R. | ANSWER: The maximum value is R, and R = 10. (6 cos x + 8 sin x = 10 cos (x - arctan(8/6)))
MCQ
Quick Quiz
What is the value of R when transforming 7 cos x + 24 sin x into R cos (x - α)?
7
24
25
31
The Correct Answer Is:
C
To find R, we use the formula R = sqrt(a^2 + b^2). Here, a = 7 and b = 24. So, R = sqrt(7^2 + 24^2) = sqrt(49 + 576) = sqrt(625) = 25.
Real World Connection
In the Real World
This concept is used in signal processing, like in your mobile phone! When your phone receives signals for calls or data, these signals are often combinations of sine and cosine waves. Engineers at companies like Jio or Airtel use this transformation to simplify these complex signals, making it easier to filter noise and extract clear information for your communication.
Key Vocabulary
Key Terms
TRANSFORMATION: Changing one mathematical form into another, usually simpler one. | AMPLITUDE (R): The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. In this context, it's the maximum value the combined function can reach. | PHASE SHIFT (α): A horizontal shift of the graph of a trigonometric function. | TRIGONOMETRIC IDENTITIES: Equations involving trigonometric functions that are true for every value of the variables. | ARCTAN: The inverse tangent function, used to find the angle when its tangent value is known.
What's Next
What to Learn Next
Next, you can explore how to transform 'a cos x + b sin x' into R sin (x + α) or R sin (x - α). Understanding these variations will give you even more tools to solve complex trigonometric problems and prepare you for higher-level mathematics in Class 11 and 12.
