Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\begin{aligned} &a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C), \text { where }\\\ &C=\arctan (b / a) \text { and } a>0 \end{aligned}$$.

Step-by-Step Solution

Answer

The given identity is verified as the terms on the left-hand side of the equation, after appropriate transformations and substitutions according to the given conditions, match exactly with the right-hand side.

Step 1: Preparation

Before comparing the two sides, we will express the right-hand side in a form that matches the left-hand side more closely. We do this by employing the formula for sine of sum of two angles.

Step 1: Express the Right-hand Side

The formula for sine of sum of two angles is: \(\sin (A + B) = \sin A \cos B + \cos A \sin B\).Applying this to the right-hand side of the original identity, we have:\(\sqrt{a^{2}+b^{2}} (\sin B\theta \cos C + \cos B\theta \sin C)\).

Step 2: Substitute for \(C\)

We are given that \(C = \arctan (b / a)\), so \(\sin C = b / \sqrt{a^{2}+b^{2}}\) and \(\cos C = a / \sqrt{a^{2}+b^{2}}\). Substituting these values in we get: \(a \sin B\theta + b \cos B\theta\).

Step 3: Comparison of Both sides

This gives us the exact same expression as on the left-hand side from the original equation, providing us proof of the identity.

Key Concepts

sine of sum formula

When dealing with trigonometry, one often encounters the sine of sum formula. This handy formula helps us find the sine of the sum of two angles, which can be particularly helpful in verifying identities or solving trigonometric equations. The formula is given by:

• \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

It involves the components of each angle separately, allowing us to express complex combinations in terms of more elementary sine and cosine functions.
For example, in the given exercise, we applied this formula to rewrite the expression \( \sqrt{a^2 + b^2} \sin(B\theta + C) \). By expanding it, we can derive expressions that are easier to compare or manipulate, like transitioning between different forms of trigonometric expressions.

arctangent function

The arctangent function, often written as \( \arctan(x) \), is the inverse operation of the tangent function. It returns the angle whose tangent is the given number. The range of \( \arctan(x) \) is typically between \(-\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians.

• It serves a crucial role in converting between ratios and angles, which is essential in trigonometric identity verification.

In the identity we're verifying, \( C = \arctan(b/a) \), we're using the arctangent to find an angle \( C \) such that the tangent of \( C \) is \( \frac{b}{a} \). This allows us to translate angle information further into sine and cosine components using their known relationships with tangent functions.

identity verification

Identity verification in trigonometry entails showing that two expressions are equivalent under all conditions defined. It's akin to proving that a mathematical statement holds true universally. Such tasks often require us to apply various trigonometric identities and manipulate expressions.
In our scenario, we used identity verification to confirm that \( a\sin(B\theta) + b\cos(B\theta) \) matches \( \sqrt{a^2 + b^2} \sin(B\theta + C) \). We strategically employed trigonometric identities, like the sine of sum formula, and substituted known values as provided.

• The process involves breaking down more complicated expressions into standardized components.
• Then, these components are compared to show equality or derive a simpler form.

trigonometric expressions

Trigonometric expressions involve variables combined with sine, cosine, tangent, and other trigonometric functions. They form the building blocks in trigonometry for modeling periodic phenomena, solving oscillation problems, and performing complex calculations.
Understanding these expressions includes being comfortable with basic trigonometric functions and how they interrelate. They also often require manipulation through identities or formulas, like:

• \( \sin(A + B) \)
• \( \cos(A + B) \)

In the verification task, understanding and properly using these expressions determined our ability to confirm the given identity. Familiarity with expressions involving sine and cosine in particular helps break down and reassemble results into desired forms, facilitating proof completion and calculation efficiency.