Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Use the even-odd identities to write of the following expressions as a trigonometric function of a positive number $$\sin (-2.5)$$

Step-by-Step Solution

Answer

\(-\sin (2.5)\)

Step 1: Recognize Even-Odd Identities

Even-Odd identities in trigonometry help simplify expressions. The sine function, \(\sin (-x) = -\sin (x)\), is an odd function, meaning it changes sign when the angle is negative.

Step 2: Apply Sine's Odd Identity

Given the expression \(\sin (-2.5)\), apply the odd identity \(\sin(-x) = -\sin(x)\). This gives us: \[\sin (-2.5) = -\sin (2.5)\]

Step 3: Express in Trigonometric Function

Now, \(-\sin (2.5)\) is expressed as a trigonometric function of a positive number, specifically \(\sin (2.5)\) with a negative sign.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in geometry and calculus, helping to explore relationships within triangles and model periodic phenomena.

• Sine, cosine, and tangent are the primary trigonometric functions, with cosecant, secant, and cotangent being their reciprocals.
• These functions relate the angles and sides of a triangle, mainly focusing on right-angled triangles.
• They extend to the unit circle representation, covering angles beyond 0 and 90 degrees, including negative angles and angles greater than 360 degrees.

To solve trigonometric problems, it's essential to understand identities, which are equations involving trigonometric functions that are true for all values of the involved variables.
Examples include the Pythagorean identity and even-odd identities. These identities allow simplifying complex expressions and solving equations efficiently.

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the most frequently used trigonometric functions. It relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse.

• In the unit circle, \( \sin(x) \) represents the y-coordinate of a point on the circle corresponding to a given angle \( x \).
• The function is periodic with a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) units.
• The sine function exhibits symmetry, being an odd function.

Being an odd function means that \( \sin(-x) = -\sin(x) \), as illustrated in our exercise. This property is critical in transforming expressions involving negative angles into a more standard format.
Understanding the behavior of the sine function in different quadrants and its properties aids in solving a wide range of mathematical problems.

Simplifying Expressions

Simplifying expressions in mathematics involves reducing them to their simplest form, making them easier to understand and solve. In the realm of trigonometry, this often involves using identities and known properties.

• Even-Odd identities help simplify expressions with negative angles. For odd functions like sine, \( \sin(-x) = -\sin(x) \), converting a negative angle to a positive one with a sign change.
• The goal is to express functions in terms of a positive angle, which can be simpler to evaluate or analyze.

By transforming \( \sin(-2.5) \) to \( -\sin(2.5) \), the expression is easier to work with, avoiding potential confusion that may arise from negative angles or unconventional formats.
Simplification also lays the groundwork for further algebraic manipulation or substitution, which are commonly needed in solving equations or optimizing functions.