Understanding Harmonic Mean, Arithmetic Mean, and Trigonometry

This problem involves understanding the concepts of Harmonic Mean (HM) and Arithmetic Mean (AM) applied to trigonometric values, specifically sin α and cos α. We are given the expressions for sin β and sin θ in terms of sin α and cos α and need to verify two trigonometric identities.

Let's start by recalling the definitions of Arithmetic Mean and Harmonic Mean for two numbers.

Definitions of Means

• Arithmetic Mean (AM): For two numbers 'a' and 'b', the AM is given by \(\frac{a+b}{2}\).
• Harmonic Mean (HM): For two numbers 'a' and 'b', the HM is given by \(\frac{2ab}{a+b}\).

According to the question:

• sin β is the harmonic mean of sin α and cos α.
• sin θ is the arithmetic mean of sin α and cos α.

Calculating sin β and sin θ

Using the definitions, we can write the expressions for sin β and sin θ in terms of sin α and cos α:

For sin β (Harmonic Mean of sin α and cos α):

\(\sin \beta = \frac{2 (\sin \alpha)(\cos \alpha)}{\sin \alpha + \cos \alpha}\)

For sin θ (Arithmetic Mean of sin α and cos α):

\(\sin \theta = \frac{\sin \alpha + \cos \alpha}{2}\)

Analyzing Statement 1: \(\sqrt 2 \sin \left( {\alpha + \frac{\pi }{4}} \right)\sin \beta = \sin 2\alpha\)

Let's evaluate the Left Hand Side (LHS) of Statement 1 using the expressions we derived.

LHS = \(\sqrt 2 \sin \left( {\alpha + \frac{\pi }{4}} \right)\sin \beta\)

First, we expand \(\sin \left( {\alpha + \frac{\pi }{4}} \right)\) using the trigonometric identity for the sine of a sum of angles: \(\sin(A+B) = \sin A \cos B + \cos A \sin B\).

\(\sin \left( {\alpha + \frac{\pi }{4}} \right) = \sin \alpha \cos \frac{\pi }{4} + \cos \alpha \sin \frac{\pi }{4}\)

We know that \(\cos \frac{\pi }{4} = \frac{1}{\sqrt 2}\) and \(\sin \frac{\pi }{4} = \frac{1}{\sqrt 2}\). Substituting these values:

\(\sin \left( {\alpha + \frac{\pi }{4}} \right) = \sin \alpha \left( \frac{1}{\sqrt 2} \right) + \cos \alpha \left( \frac{1}{\sqrt 2} \right) = \frac{1}{\sqrt 2} (\sin \alpha + \cos \alpha)\)

Now substitute this result and the expression for sin β back into the LHS expression of Statement 1:

LHS = \(\sqrt 2 \left[ \frac{1}{\sqrt 2} (\sin \alpha + \cos \alpha) \right] \left[ \frac{2 \sin \alpha \cos \alpha}{\sin \alpha + \cos \alpha} \right]\)

We can cancel out the \(\sqrt 2\) terms and also the \((\sin \alpha + \cos \alpha)\) terms from the numerator and denominator, assuming \(\sin \alpha + \cos \alpha \neq 0\).

LHS = \((\sin \alpha + \cos \alpha) \left[ \frac{2 \sin \alpha \cos \alpha}{\sin \alpha + \cos \alpha} \right] = 2 \sin \alpha \cos \alpha\)

Using the double angle identity for sine, \(\sin 2\alpha = 2 \sin \alpha \cos \alpha\):

LHS = \(\sin 2\alpha\)

This is equal to the Right Hand Side (RHS) of Statement 1. Thus, Statement 1 is correct.

Analyzing Statement 2: \(\sqrt 2 \sin \theta = \cos \left( {\alpha - \frac{\pi }{4}} \right)\)

Let's evaluate the Left Hand Side (LHS) of Statement 2.

LHS = \(\sqrt 2 \sin \theta\)

Substitute the expression for sin θ that we found:

LHS = \(\sqrt 2 \left[ \frac{\sin \alpha + \cos \alpha}{2} \right]\)

We can simplify \(\frac{\sqrt 2}{2}\) to \(\frac{1}{\sqrt 2}\):

LHS = \(\frac{1}{\sqrt 2} (\sin \alpha + \cos \alpha)\)

Now let's evaluate the Right Hand Side (RHS) of Statement 2.

RHS = \(\cos \left( {\alpha - \frac{\pi }{4}} \right)\)

Expand \(\cos \left( {\alpha - \frac{\pi }{4}} \right)\) using the trigonometric identity for the cosine of a difference of angles: \(\cos(A-B) = \cos A \cos B + \sin A \sin B\).

\(\cos \left( {\alpha - \frac{\pi }{4}} \right) = \cos \alpha \cos \frac{\pi }{4} + \sin \alpha \sin \frac{\pi }{4}\)

Using the values \(\cos \frac{\pi }{4} = \frac{1}{\sqrt 2}\) and \(\sin \frac{\pi }{4} = \frac{1}{\sqrt 2}\):

\(\cos \left( {\alpha - \frac{\pi }{4}} \right) = \cos \alpha \left( \frac{1}{\sqrt 2} \right) + \sin \alpha \left( \frac{1}{\sqrt 2} \right) = \frac{1}{\sqrt 2} (\cos \alpha + \sin \alpha)\)

Since addition is commutative, \((\cos \alpha + \sin \alpha)\) is the same as \((\sin \alpha + \cos \alpha)\).

RHS = \(\frac{1}{\sqrt 2} (\sin \alpha + \cos \alpha)\)

Comparing the LHS and RHS of Statement 2:

LHS = \(\frac{1}{\sqrt 2} (\sin \alpha + \cos \alpha)\)

RHS = \(\frac{1}{\sqrt 2} (\sin \alpha + \cos \alpha)\)

Since LHS = RHS, Statement 2 is correct.

Conclusion: Verifying Trigonometric Identities

Both Statement 1 and Statement 2 have been verified to be correct based on the definitions of harmonic and arithmetic means and fundamental trigonometric identities.

Revision Table: Key Concepts in Trigonometry and Means

Additional Information on Means and Trigonometric Formulas

Understanding different types of means (Arithmetic, Geometric, Harmonic) is useful in various mathematical contexts. The relationship AM \(\ge\) GM \(\ge\) HM (for positive numbers) is a fundamental inequality.

Trigonometric identities are equations that are true for all values of the variables involved. They are essential tools for simplifying expressions and solving trigonometric equations. Common identities include sum and difference formulas, double-angle formulas, half-angle formulas, product-to-sum, and sum-to-product identities.

Problems combining concepts from different areas of mathematics, like means and trigonometry, are common. Breaking down such problems by first understanding the definitions and then applying relevant formulas step-by-step is a good approach.