Question

Statement 1 If \(\mathrm{r}_{1}, \mathrm{r}_{2}, \mathrm{r}_{3}\) are in \(\mathrm{HP}, \sin \mathrm{A}, \operatorname{sinB}, \sin \mathrm{C}\) are in \(\mathrm{HP}\). and Statement 2 If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in \(\mathrm{HP} \frac{1}{\mathrm{a}}, \frac{1}{\mathrm{~b}}, \frac{1}{\mathrm{c}}\) are in \(\mathrm{AP}\).

Short answer

Are 1/a, 1/b, and 1/c in Arithmetic Progression when a, b, c are in Harmonic Progression? Answer: Statement 1 (sinA, sinB, sinC in HP when r1, r2, r3 are in HP) is not necessarily true, as we cannot derive a relationship between sinA, sinB, sinC and r1, r2, r3 directly. Statement 2 (1/a, 1/b, and 1/c are in AP when a, b, c are in HP) is true, as the differences between consecutive terms are constant when a, b, c are in HP.

Step-by-step solution

Understand the concept of Harmonic Progression

In a Harmonic Progression, the reciprocals of the terms are in Arithmetic Progression. So, if r1, r2, r3 are in HP, then 1/r1, 1/r2, and 1/r3 are in AP.

Convert sinA, sinB, sinC in terms of r1, r2, r3

Using the sine rule in trigonometry, we can write sinA, sinB, and sinC in terms of r1, r2, and r3. The sine rule states: \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R\) where a, b, c are the sides of the triangle and R is the circumradius. (Note: We are given r1, r2, and r3 are the exradii of the triangle which are related to R by the formula \(r_i=\frac{R\cdot s-a_i}{s}\), but sinA, sinB, and sinC are not directly related to them, so we cannot derive the required relationship between sinA, sinB, and sinC. Hence, it is not possible to prove the given statement directly.)

Conclude the analysis

Since we cannot derive a relationship between sinA, sinB, sinC and r1, r2, r3, we cannot show that sinA, sinB, sinC are in HP when r1, r2, r3 are in HP. So, Statement 1 is not necessarily true. Statement 2:

Recall the relationship between AP and HP

If terms a, b, c are in HP, then the reciprocals of the terms, 1/a, 1/b, and 1/c, are in AP.

Verify the relationship

To show that 1/a, 1/b, and 1/c are in AP, we need to check if the differences between consecutive terms are constant. Let's calculate these differences: \(\frac{1}{b} - \frac{1}{a} = \frac{a-b}{ab}\) \(\frac{1}{c} - \frac{1}{b} = \frac{b-c}{bc}\) Now, let's compare these differences: \(\frac{a-b}{ab} = \frac{b-c}{bc}\)

Simplify and conclude

Cross multiply to get: \((a-b)bc = (b-c)ab\) As we can see, this relationship holds true. Therefore, we can conclude that if a, b, c are in HP, then 1/a, 1/b, and 1/c are in AP. So, Statement 2 is true.

Key concepts

Arithmetic Progression

Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two successive terms is always the same. This consistent difference is called the common difference.
For example, in the sequence 2, 4, 6, 8, the common difference is 2.

• The general form of an arithmetic sequence can be written as: \(a, a+d, a+2d, a+3d,\) and so on, where \(a\) is the first term and \(d\) is the common difference.
• This form can help you find any term in the sequence if you know the first term and the common difference. Simply use the formula: \(a_n = a + (n-1)d\)

Understanding how to recognize AP is crucial when dealing with harmonic progressions, where the reciprocals of the terms of a harmonic progression form an arithmetic progression.

Sine Rule

The Sine Rule is a fundamental concept in trigonometry that relates the sides of a triangle to the sines of its angles. It states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \] where \(a, b, c\) are the lengths of the sides opposite angles \(A, B, C\) respectively, and \(R\) is the circumradius of the triangle.

• It is particularly useful for solving triangles, allowing you to find unknown sides or angles when given other specific details.
• The rule applies to any type of triangle, making it a versatile tool in trigonometry.

In exercise problems, using the Sine Rule helps connect angles and sides to broader concepts like the circumradius, facilitating comprehensive solutions.

Trigonometry

Trigonometry is the branch of mathematics that studies the relationships involving lengths and angles of triangles. Its principles are used extensively in various fields including physics, engineering, and computer science.

• The core functions in trigonometry are sine, cosine, and tangent, which relate angles of a right triangle to the lengths of its sides.
• Understanding these functions allows for the solving of complex problems, including those involving non-right triangles when combined with the sine and cosine rules.

In geometric contexts, trigonometric principles often help derive properties like areas, heights, and even volumes, enriching our understanding of spatial structures.

Exradii

In a triangle, the exradius is the radius of an excircle, which is a circle tangent to one side of the triangle and the extensions of the other two sides. A triangle has three excircles, each associated with one of the triangle's vertices.

• The radius of an excircle opposite a side \(a\) is often denoted as \(r_a\), calculated by the formula \( r_a = \frac{A}{s-a}\), where \(A\) is the area of the triangle and \(s\) is the semi-perimeter \( s = \frac{a+b+c}{2} \).
• These exradii are essential in computations involving circle-related properties like the circumradius and the inradius.

Understanding exradii can help in solving complex geometric problems, as they contextualize relationships between a triangle's sides and its circumscribed and inscribed circles.

Circumradius

The circumradius of a triangle is the radius of the circumcircle, which passes through all three vertices of the triangle. This radius is significant because it ties angular and side data of a triangle into a cohesive whole.

• The formula used to calculate the circumradius \(R\) is \( R = \frac{abc}{4A} \), where \(a, b, c\) are the side lengths of the triangle and \(A\) is the area.
• This measurement helps in expressing various trigonometric rules, such as the sine rule, which connects angle measures to this fundamental circle-related quantity.

By mastering the concept of circumradius, one can effectively tackle problems that involve cyclic quadrilaterals and other complex figures that build upon basic triangular properties.