Question

In a triangle, angles \(\mathrm{A}, \mathrm{B}, \mathrm{C}\), are in AP. Then \(\lim _{\mathrm{A} \rightarrow \mathrm{C}} \frac{\sqrt{3-4 \sin \mathrm{A} \sin \mathrm{C}}}{|\mathrm{A}-\mathrm{C}|}\) is (a) 1 (b) 2 (c) 3 (d) 4

Short answer

Answer: The limit of the given expression as A approaches C is -$\frac{\sqrt{3}}{2}$.

Step-by-step solution

Write angles A, B, and C in terms of Arithmetic Progression

An arithmetic progression has the form \(a, a + d, a + 2d,\) where \(a\) is the first term and \(d\) is the common difference. In this case, we can write: A = a B = a + d C = a + 2d

Use angle sum property of triangle

The sum of angles in a triangle is 180 degrees or \(\pi\) radians. So, we can write: A + B + C = \(\pi\) a + (a + d) + (a + 2d) = \(\pi\) 3a + 3d = \(\pi\) a + d = \(\frac{\pi}{3}\)

Replace C with A

We want to find the limit as A approaches C. Replace C with A in the previous result. A + B + A = \(\pi\) 2A + B = \(\pi\) Now let's use the expression for B from Step 1: 2A + a + d = \(\pi\) But we know from Step 2 that a+d = \(\frac{\pi}{3}\), so 2A + \(\frac{\pi}{3}\) = \(\pi\)

Rewrite the given expression

Now, we can solve for \(\sin{C}\) in terms of A from Step 3: \(2A = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\) \(A = \frac{\pi}{3}\) Substitute this value into the given expression: \(\lim_{A \rightarrow \frac{\pi}{3}} \frac{\sqrt{3 - 4\sin{A}\sin{C}}}{|A - C|}\)

Apply L'Hopital's rule

Let \(f(A) = \sqrt{3 - 4\sin{A}\sin{C}}\) and \(g(A) = |A - C|\). To apply L'Hopital's Rule, we need to find \(\lim_{A \rightarrow C} \frac{f'(A)}{g'(A)}\). \(f'(A) = \frac{-4\cos{A}\sin{C}}{2\sqrt{3 - 4\sin{A}\sin{C}}}\) \(g'(A) = 1\) So we have: \(\lim_{A \rightarrow \frac{\pi}{3}} \frac{-4\cos{A}\sin{C}}{2\sqrt{3 - 4\sin{A}\sin{C}}} = -2\cos{\frac{\pi}{3}}\sin{C}\) From Step 4, we have: \(A = \frac{\pi}{3}\) \(\lim_{A \rightarrow C} \frac{f'(A)}{g'(A)} = -2\cos{\frac{\pi}{3}}\sin{\frac{\pi}{3}} = -2\cdot\frac{1}{2}\cdot\frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}\) Thus, the limit of the given expression as \(\lim_{A \rightarrow C} \frac{\sqrt{3 - 4\sin{A}\sin{C}}}{|A - C|}\) does not match with any of the given answer choices (a), (b), (c), or (d).

Key concepts

Arithmetic Progression

Arithmetic progression is a sequence of numbers in which the difference of any two successive members is a constant. This sequence is very useful in mathematics, especially when dealing with series and patterns.
For example, if you have three angles in a triangle forming an arithmetic progression, they can be expressed as:

• First angle: \(a\) (let's call this \(A\))
• Second angle: \(a + d\) (call this \(B\))
• Third angle: \(a + 2d\) (and this \(C\))

Here, \(a\) is the first term and \(d\) is the common difference between the terms.
This is particularly useful in triangles, where the sum of angles must equal 180 degrees or \(\pi\) radians. So, if we use the arithmetic progression form, the equation becomes: \[ A + B + C = \pi \] In our example: \[ a + (a + d) + (a + 2d) = \pi \] Simplifying gives: \[ 3a + 3d = \pi \] Which reduces to: \[ a + d = \frac{\pi}{3} \] This simplifies solving problems involving angles in arithmetic progression in a triangle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides are defined. These identities simplify complex expressions, making calculations easier.
For example, the angles and their respective sine values are used frequently in problems involving triangles and circles. These identities help in expressing angles and their sines in more manageable forms. Some important basic identities include:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \sin{(A + B)} = \sin A \cos B + \cos A \sin B \)

In our problem, we use the identities to express the trigonometric function \(\sin{C}\) in terms of known angles. By manipulating these identities, we can express \(\sqrt{3 - 4\sin{A}\sin{C}}\) in a simpler form.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for finding the limit of an indeterminate form, like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In calculus problems involving limits, if directly substituting the point into the function gives an indeterminate form, L'Hôpital's rule allows us to differentiate the numerator and the denominator separately and then take the limit.
To apply L'Hôpital's Rule, you should:

• Check if the direct substitution results in an indeterminate form.
• Differentiate the numerator and the denominator of the function.
• Take the limit of the resulting function.

In our exercise, we utilize L'Hôpital's rule on the expression \(\lim_{A \rightarrow C} \frac{\sqrt{3 - 4\sin{A}\sin{C}}}{|A - C|}\). Here, differentiating both the top and bottom allowed us to simplify and find the value of the limit. This method is key for handling complex limit problems involving derivatives when the direct approach is impractical.