Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-16: Express as a trigonometric function of one angle. $$ \cos 10^{\circ} \sin 5^{\circ}-\sin 10^{\circ} \cos 5^{\circ} $$

Step-by-Step Solution

Answer

\(\sin(5^\circ)\)

Step 1: Recognize the Trigonometric Identity

We start by recognizing that the expression \(\cos 10^{\circ} \sin 5^{\circ} - \sin 10^{\circ} \cos 5^{\circ}\) resembles the sine of a difference formula, which is given by the identity \(\sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B)\).

Step 2: Identify Corresponding Angles

By comparing \(\cos(10^\circ) \sin(5^\circ) - \sin(10^\circ) \cos(5^\circ)\) with \(\sin(A) \cos(B) - \cos(A) \sin(B)\), we identify \(A = 10^\circ\) and \(B = 5^\circ\).

Step 3: Apply the Sine Difference Identity

Using the identity \(\sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B)\), we substitute the angles: \(\sin(10^\circ - 5^\circ)\).

Step 4: Simplify the Expression

Calculate \(10^\circ - 5^\circ\) which equals \(5^\circ\). Thus, the expression becomes \(\sin(5^\circ)\).

Key Concepts

Sine Difference Identity

The sine difference identity is a crucial tool in trigonometry that helps simplify expressions involving trigonometric functions. It states: \[\sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B)\]This identity enables us to express the sine of the difference between two angles as a combination of their sine and cosine values. Such identities are very helpful when working with more complex trigonometric equations, and they provide an alternative way to simplify and solve problems.

• The identity makes it more straightforward to handle operations involving subtraction of angles.
• The formula is particularly useful in transforming a complicated expression into a simpler one, as seen when reducing \(\cos 10^{\circ} \sin 5^{\circ} - \sin 10^{\circ} \cos 5^{\circ}\) to \(\sin(10^{\circ} - 5^{\circ})\).
• Recognizing these patterns in calculations can save time and reduce error chances.

Knowing when and how to apply these identities vastly increases efficiency in solving trigonometric equations.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and others, are fundamental in understanding how angles and measurements relate in the world of mathematics, especially in geometrical contexts. Each function is derived from the properties of triangles—mainly right triangles—and applies to a wide range of problems.

• Sine: Represents the opposite side over the hypotenuse in right triangles.
• Cosine: Represents the adjacent side over the hypotenuse.
• Trigonometric functions help convert angles into other measurable phenomena, including sound waves, light, and even heat patterns.

These functions go beyond mere triangle measurements; they are tools for expressing wave patterns, periodic events, and much more in both practical and theoretical applications.
Each trigonometric function has specific properties and behavior according to the angle's quadrant in a unit circle, making them versatile in mathematical analysis.

Angles in Trigonometry

Angles are a foundational aspect of trigonometry, representing the rotation from one ray to another around a shared vertex. They are typically measured in degrees or radians, with full rotation being \(360^{\circ}\) or \(2\pi\) radians. Angles provide the basis for defining the trigonometric functions.

• Understanding angles is key to exploring the relationships and functions they form in trigonometry.
• Angles determine the sign and value of trigonometric functions.
  The quadrant in which an angle lies affects whether a trigonometric value is positive or negative.
• In problems involving identities like the sine difference identity, it's crucial to accurately determine and work with the given angles.

By accurately calculating and manipulating angles, mathematicians and scientists can model and solve a wide array of practical problems related to waves, cycles, rotations, and more. Recognizing the incremental steps in these calculations is essential for achieving precision and unlocking advanced topics within the subject.