Problem 104 - Precalculus Mathematics for Calculus

Question

If you start with a trigonometric expression and rewrite it or simplify it, then setting the original expression equal to the rewritten expression yields a trigonometric identity. For instance, from Example 1 we get the identity $$\cos t+\tan t \sin t=\sec t$$ Use this technique to make up your own identity, then give it to a classmate to verify.

Step-by-Step Solution

Answer

\( \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + \sin 2x \).

Step 1: Choose a Trigonometric Expression

Let's start with a known trigonometric function. We choose the expression \( \sin^2 x + \cos^2 x \). This is a common trigonometric identity that simplifies to \( 1 \).

Step 2: Introduce a New Component

To create a new identity, let's introduce a new component to the existing expression. We can consider adding \( 2\sin x \cos x \) to our chosen expression: \( \sin^2 x + \cos^2 x + 2\sin x \cos x \).

Step 3: Simplify the New Expression

Now, simplify the expression \( \sin^2 x + \cos^2 x + 2\sin x \cos x \). The first two terms, \( \sin^2 x + \cos^2 x \), simplify to \( 1 \). The term \( 2\sin x \cos x \) can be rewritten using the double angle identity for sine, which is \( \sin 2x \). So the expression simplifies to \( 1 + \sin 2x \).

Step 4: Set the Original Expression Equal to the New Expression

Now we set the original trigonometric expression equal to the simplified expression. Thus, \( \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + \sin 2x \).

Step 5: Verify the Identity

To verify the identity, substitute arbitrary values for \( x \). For instance, if \( x = 0 \), the left-hand side becomes \( \sin^2 (0) + \cos^2 (0) + 2\sin (0)\cos (0) = 1 \), and the right-hand side is \( 1 + \sin (0) = 1 \). The identity holds for this value of \( x \) and can be checked similarly for other values.

Key Concepts

Trigonometric Expressions

Trigonometric expressions are mathematical phrases that involve the basic trigonometric functions: sine, cosine, tangent, and their reciprocals. These expressions can take many forms, combining sums, products, or powers of trigonometric functions. For instance, an expression might look like \(\sin x + \cos x\) or \(\tan x\cdot \sec x\).
Understanding these expressions is crucial because it allows us to simplify complex problems, proving useful in many areas of mathematics and applied sciences. Trigonometric expressions often appear in problems relating to waves, circles, and oscillations, making mastery of their properties very beneficial.

• Trigonometric functions can be rearranged and combined to transform the expressions.
• They can be paired with algebraic operations to form new expressions.
• Expressions are motive parts in forming and verifying trigonometric identities.

Double Angle Identities

Double angle identities are special formulas in trigonometry that simplify complex expressions involving angles twice the size of a given angle. For sine and cosine, these identities are particularly useful:

• \( \sin 2x = 2\sin x\cos x \)
• \( \cos 2x = \cos^2 x - \sin^2 x \)
• Alternatively, \( \cos 2x \) can also be expressed as \( 1 - 2\sin^2 x \) or \( 2\cos^2 x - 1 \).

These identities enable the transformation of trigonometric expressions involving double angles into simpler forms, often reducing the complexity of the problem. For instance, in this exercise, the identity for \( \sin 2x = 2\sin x\cos x \) is used to simplify the given expression. The double angle identities are widely applied in advanced mathematics, physics, and engineering problems.

Simplifying Expressions

Simplifying expressions helps in transforming a complex expression into a more manageable form. In trigonometry, simplifying expressions often means using identities to combine terms and reduce them down:

• Identify common trigonometric identities that appear within the expression.
• Use equivalent expressions to replace more complex terms.
• Constantly verify if the expression can be reduced further.

In the given exercise, the expression \( \sin^2 x + \cos^2 x \ + 2\sin x\cos x \) is simplified by recognizing \( \sin^2 x + \cos^2 x = 1 \) and expressing \( 2\sin x\cos x \) as \( \sin 2x \). This results in the cleaner expression \( 1 + \sin 2x \). Simplifying helps in making computations more straightforward and ensures that the trigonometric identities can be easily verified.

Verifying Identities

Verifying identities in trigonometry means proving that two expressions, usually an original and a rewritten form, are equivalent for all values of the variable involved. This process is critical because it confirms that the transformation of an expression is valid, maintaining its mathematical integrity.

• Select specific values of the variable to test the equivalence.
• Both sides of the expression should yield the same result.
• Check for several different values to ensure that the identity holds firm across its domain.

In the solution, you substitute different values for \( x \) and see if both the original and simplified sides of the expression maintain equality. For example, if \( x = 0 \), both sides equate to 1, confirming part of the identity. Verifying identities is like a mathematical double-check, ensuring that manipulations of the trigonometric expressions wire correctly and proving their truth unequivocally.