Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the expression as the sine, cosine, or tangent of an angle. $$\sin 60^{\circ} \cos 10^{\circ}-\cos 60^{\circ} \sin 10^{\circ}$$

Step-by-Step Solution

Answer

The expression \(\sin 60^{\circ} \cos 10^{\circ}-\cos 60^{\circ} \sin 10^{\circ}\) can be written as \(\sin(50^{\circ})\).

Step 1: Identify the formula

Notice that the given expression matches the format of the sine subtraction formula: sin(a - b) = sin(a)cos(b) - cos(a)sin(b). Here, a = 60° and b = 10°.

Step 2: Apply the formula

Correspond the given expression to the sine subtraction formula. This means that the expression can be rewritten as sin(60° - 10°).

Step 3: Simplify the expression

Perform the subtraction to simplify the expression. That gives you sin(50°).

Key Concepts

sine subtraction formula

The sine subtraction formula is a powerful tool in trigonometry used to simplify expressions involving sine and cosine. It is represented as \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \). This formula is especially useful when you need to break down complex trigonometric expressions into simpler terms.
Think of it as a "recipe" for creating a new sine function from two angles, \( a \) and \( b \). The sine subtraction formula can help you transform an expression that initially looks complex into a simpler form, or even a single trigonometric function.
In our example, \( \sin(60^{\circ})\cos(10^{\circ}) - \cos(60^{\circ})\sin(10^{\circ}) \) perfectly aligns with this formula. By recognizing the pattern, you see it as \( \sin(60^{\circ} - 10^{\circ}) \), which makes the calculation straightforward. This highlights the elegance and efficiency of using trigonometric identities in problem-solving.

trigonometric expression simplification

Trigonometric expression simplification involves rewriting a complex trigonometric expression into a simpler, often more manageable form. This is not just about making the expression shorter, but also about revealing insights into its properties or making calculations more straightforward.
A common technique involves recognizing and applying trigonometric identities, like the sine subtraction formula. By identifying patterns within the expression, you can often substitute parts of it with simpler equivalents. This is part of what makes trigonometry both challenging and rewarding.
In our case, the original expression \( \sin(60^{\circ})\cos(10^{\circ}) - \cos(60^{\circ})\sin(10^{\circ}) \) was simplified to \( \sin(50^{\circ}) \) by applying the sine subtraction formula. Through simplification, problems that initially appear daunting become accessible, allowing you to compute or further manipulate the result with little effort.

angle subtraction in trigonometry

Angle subtraction in trigonometry is a technique where you compute the sine, cosine, or tangent of the difference between two angles. This often involves using subtraction formulas like the sine subtraction formula. These formulas provide a way to express complex expressions in simpler terms.
For any two given angles, finding the trigonometric function of their difference can reveal interesting properties and relationships. By understanding how subtraction affects these functions, you gain deeper insights into how angles interact within a trigonometric context.
In our example, the angle subtraction \( \sin(60^{\circ} - 10^{\circ}) \) resulted in \( \sin(50^{\circ}) \). Calculating the trigonometric function of \( 50^{\circ} \), you effectively find the outcome of a complex expression in a clean and efficient manner. This showcases the power of angle subtraction and serves as a reminder of the interconnectedness within trigonometry.