Problem 49 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Verify each identity. $$ \frac{\cos (x+h)-\cos x}{h}=\cos x \frac{\cos h-1}{h}-\sin x \frac{\sin h}{h} $$

Step-by-Step Solution

Answer

The given identity has been verified after substitution and simplification using trigonometric sum identities.

Step 1: Expand Cosine of Sum

Start by expanding the cosine of the sum in the numerator using the sum formula. We have: \(\cos (x+h)= \cos x \cos h - \sin x \sin h\)

Step 2: Substitute into the Identity

We substitute this back into the original identity to have: \(\frac{\cos (x+h)-\cos x}{h} = \frac{(\cos x \cos h - \sin x \sin h) - \cos x}{h}\). Simplify to get: \(= \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h}\)

Step 3: Rewrite as original identity

This is equivalent to the original identity, hence the given identity is verified.

Key Concepts

Sum and Difference Formulas

Sum and difference formulas are essential tools in trigonometry, helping us break down complex trigonometric expressions into simpler ones. These formulas are vital for manipulating angles and solving various trigonometric equations or identities. The sum formula for cosine, used in the exercise, is one of the most commonly applied.

• The formula for the sum of two angles for cosine is: \[\cos(a + b) = \cos a \cos b - \sin a \sin b\]
• This helps express the cosine of the sum of two angles as a product of sines and cosines.
• The formula simplifies calculations by expressing a challenging term in terms of known values.

These identities enable you to verify the correctness of expressions, like in the problem, by rearranging and substituting terms accurately. With practice, mastering these identities makes solving trigonometric equations more intuitive and manageable.

Cosine of a Sum

The cosine of a sum formula is a powerful tool in trigonometry that allows us to transform an expression involving a sum of angles into one that involves only products, making it simpler to evaluate and manipulate. This formula is represented as:\[\cos(x + h) = \cos x \cos h - \sin x \sin h\]

• This relation is key when trying to find the cosine of an angle expressed as a sum.
• By expressing it in terms of basic trigonometric functions, we can handle more complex expressions.
• In practice, expanding expressions using the cosine sum formula simplifies them for easy calculation.

In solving trigonometric identities or equations, the ability to expand and re-arrange terms using this formula often reveals the identity that needs verification or simplifies the expression.

Verifying Identities

Verifying identities in trigonometry means proving that two different-looking expressions are indeed the same. This involves algebraically manipulating one or both sides of the equation to transform them into each other.

• Identities involve equalities valid for all values of the involved variables.
• Steps often include substituting known identities or rewriting terms using fundamental trigonometric formulas.
• With the expansion of terms, simplifying becomes crucial to reach the final verified identity.

In the original step-by-step solution, the identity was verified by substituting the cosine of a sum and then simplifying it to match the given identity, illustrating the power of precise substitution and simplification.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are fundamental in understanding various mathematical contexts, including geometry and calculus. These functions describe relationships between the angles and sides of triangles and extend into periodic functions useful for modeling real-world phenomena.

• These include six basic functions: sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).
• Their relationships with each other are defined by various identities, such as the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\).
• Understanding these relationships helps in solving complex problems where angles and distances are involved.

In the exercise, using trigonometric identities correctly required understanding how these trigonometric functions operate and how they transform through different identities.