Question

If the angles \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) of a triangle are in an arithmetic progression and if \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) denote the lengths of the sides opposite to \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) respectively, then the value of the expression \(\frac{\mathrm{a}}{\mathrm{c}} \sin 2 \mathrm{C}+\frac{\mathrm{c}}{\mathrm{a}} \sin 2 \mathrm{~A}\) is A) \(\frac{1}{2}\) B) \(\frac{\sqrt{3}}{2}\) C) 1 D) \(\sqrt{3}\)

Short answer

C) 1

Step-by-step solution

Understand the Relationship Between the Angles

Since the angles A, B, and C are in an arithmetic progression, we can express them as B = A + d, C = A + 2d, where d is the common difference.

Use the Fact that the Sum of Angles in a Triangle is 180 Degrees

Knowing that the sum of the angles in a triangle is always 180 degrees, set up the equation A + (A+d) + (A+2d) = 180 and solve for A and d.

Solve the Equation to Find A and d

Combine like terms to get 3A+3d = 180. Divide both sides by 3 to find A + d = 60, meaning that A = 60 - d and C = 60 + d.

Express the Sine Functions in Terms of A and C

Since C = 60 + d, express sin 2C in terms of sin(120 + 2d), using the angle sum identity for sine. Do similarly for sin 2A.

Calculate sin 2A and sin 2C

Using sin 2A = sin(2(60 - d)) and sin 2C = sin(2(60 + d)), apply the sine of sum and difference identities to find the exact values.

Substitute the Sine Values in the Expression

Substitute sin 2A and sin 2C back into the expression \(\frac{a}{c} \sin 2C+\frac{c}{a} \sin 2A\) and simplify.

Use the Law of Sines

Apply the law of sines, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), to find the ratios \(\frac{a}{c}\) and \(\frac{c}{a}\).

Simplify the Expression

Substitute the ratios from the Law of Sines into the original expression. Upon simplifying, you will find that the expression equals 1.

Key concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term after the first is formed by adding a constant, called the common difference, to the previous term. In our case, the angles A, B, and C of a triangle are in such a progression. With B being the middle angle, it can be expressed as B = A + d, and the largest angle C as C = A + 2d.

This definition is crucial because it establishes a relationship between the triangle's angles that we can use to deduce the value of A and d. By knowing the sum of the angles in a triangle is always 180 degrees, we can write the equation A + (A + d) + (A + 2d) = 180 and solve for the individual angles.

Law of Sines

The Law of Sines is a critical rule in trigonometry that relates the lengths of sides of a triangle to the sines of its angles. It states that the ratio of the length of a side to the sine of the angle opposite that side is the same for all three sides of the triangle. Mathematically, it is denoted as \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

This formula plays a key role in the solution, as it allows us to find ratios between the sides based on the angles. When we later substitute these ratios into our expression \(\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A\), they simplify the problem considerably and enable us to find the correct value of the expression.

Angle Sum Identity

The angle sum identity is an important concept in trigonometry that deals with the sine or cosine of the sum or difference of two angles. For example, the sine of the sum of two angles A and B can be written as \(\sin (A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B\).

This identity assists in evaluating the sine of any angle that is not readily available in the list of standard angles for trigonometric functions. It also helps us in the current problem because by knowing the relationship between the angles, we can express \(\sin 2C\) and \(\sin 2A\) in terms of \(\sin (120 + 2d)\) and \(\sin (120 - 2d)\), simplifying our work to derive the sine values needed for the expression.

Sine Function

The sine function is one of the fundamental trigonometric functions and relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. In the unit circle, it represents the y-coordinate of the point where the terminal side of the angle intersects the circle.

Understanding the sine function is crucial for solving trigonometric problems that involve right triangles. In our exercise, we use the sine function to find \(\sin 2A\) and \(\sin 2C\), which are necessary to compute the value of the given expression. By using the identities and laws we've discussed, the computation of the sine function becomes more accessible and leads us towards the solution.