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:

\(f(\sin\theta) = -(3\sin\theta - 4\sin^3\theta)\)

\(f(\sin\theta) = -\sin(3\theta)\)

Thus, \(f(\sin\theta)\) is equal to \(-\sin3\theta\).

Revision Table - Function and Trigonometry

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.