If f(x) = x(4x2 - 3), then what is f(sinθ) equal to ?
-sin3θ
Evaluating a Function with Trigonometric Input
The problem asks us to find the value of the function f(x) when the input x is equal to sinθ. The given function is f(x) = x(4x<sup>2</sup> - 3).
Step-by-Step Solution
We are given the function:
\(f(x) = x(4x^2 - 3)\)
We need to find \(f(\sin\theta)\). To do this, we substitute \(x = \sin\theta\) into the function:
\(f(\sin\theta) = \sin\theta (4(\sin\theta)^2 - 3)\)
Simplify the expression:
\(f(\sin\theta) = \sin\theta (4\sin^2\theta - 3)\)
Now, let's expand the expression:
\(f(\sin\theta) = 4\sin^3\theta - 3\sin\theta\)
We need to compare this expression with the given options. Let's recall the triple angle formula for sine:
\(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\)
Comparing our expression \(4\sin^3\theta - 3\sin\theta\) with the formula for \(\sin(3\theta)\), we can see that our expression is the negative of the formula for \(\sin(3\theta)\).
\(4\sin^3\theta - 3\sin\theta = -(3\sin\theta - 4\sin^3\theta)\)
\(4\sin^3\theta - 3\sin\theta = -(\sin(3\theta))\)
Therefore,
\(f(\sin\theta) = -\sin(3\theta)\)
Explanation of Key Concepts
Function Evaluation
Function evaluation involves substituting a specific value or expression for the independent variable (usually \(x\)) in a function's formula and calculating the resulting value. In this case, we substituted \(\sin\theta\) for \(x\) in \(f(x)\).
Trigonometric Identities
Trigonometric identities are equations that are true for all possible values of the variables involved. The triple angle formula for sine is a key identity used here.
- Triple angle formula for sine: \(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\)
Summary of Calculation
Starting with \(f(x) = x(4x^2 - 3)\), we substitute \(x = \sin\theta\):
\(f(\sin\theta) = \sin\theta(4\sin^2\theta - 3)\)
\(f(\sin\theta) = 4\sin^3\theta - 3\sin\theta\)
Recognizing the pattern related to the triple angle formula for sine:
\(f(\sin\theta) = -(3\sin\theta - 4\sin^3\theta)\)
\(f(\sin\theta) = -\sin(3\theta)\)
| Input \(x\) | Function \(f(x)\) | Result |
|---|---|---|
| \(\sin\theta\) | \(x(4x^2 - 3)\) | \(\sin\theta(4\sin^2\theta - 3)\) |
| Simplified Expression | \(4\sin^3\theta - 3\sin\theta\) | |
| Using \(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\) | \(-\sin(3\theta)\) |
Thus, \(f(\sin\theta)\) is equal to \(-\sin3\theta\).
Revision Table - Function and Trigonometry
| Concept | Description | Example/Formula |
|---|---|---|
| Function Evaluation | Finding the value of a function for a specific input. | If \(g(x) = 2x+1\), \(g(3) = 2(3)+1 = 7\). |
| Trigonometric Identity | An equation involving trigonometric functions that is true for all valid inputs. | \(\sin^2\theta + \cos^2\theta = 1\) |
| Triple Angle Sine Formula | Identity for \(\sin(3\theta)\). | \(\sin(3\theta) = 3\sin\theta - 4\sin^3\theta\) |
Additional Information - Related Trigonometric Identities
Besides the triple angle formula for sine, there are other useful multiple angle identities:
- Double Angle Formula for Sine: \(\sin(2\theta) = 2\sin\theta\cos\theta\)
- Double Angle Formula for Cosine: \(\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\)
- Double Angle Formula for Tangent: \(\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\)
- Triple Angle Formula for Cosine: \(\cos(3\theta) = 4\cos^3\theta - 3\cos\theta\)
These identities are frequently used to simplify trigonometric expressions and solve trigonometric equations.
Important Questions from Relations and Functions
Consider the following statements:
1. The relation f defined by \(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\) is a function.
2. The relation g defined by \(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\) is a function.
Which of the statements given above is/are correct?
A function satisfies \(f(x-y)=\frac{f(x)}{f(y)}\), where f(y) ≠ 0. If f(1) = 0.5, then what is f(2) + f(3) + f(4) + f(5) + f(6) equal to ?
A mapping f : A → B defined as \(f(x)=\frac{2 x+3}{3 x+5}, x \in A\) If f is to be onto, then what are A and B equal to ?
Let R be a relation on the set N of natural numbers defined by ‘nRm ⟺ n is a factor of m’. Then which one of the following is correct?
f(xy) = f(x) + f(y) is true for all
