Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the expression. $$\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}$$

Step-by-Step Solution

Answer

The exact value of the expression is \( \sqrt{3}/2 \)

Step 1: Recognize the identity

The given expression \( \sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ} \) matches the form of the trigonometric identity \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

Step 2: Apply the identity

In this expression, \(a = 120^{\circ}\) and \(b = 60^{\circ}\). Therefore, the expression is equivalent to \( \sin(120^{\circ}-60^{\circ}) \). This simplifies to \( \sin(60^{\circ}) \).

Step 3: Use Trigonometric Values

By recalling the specific values of trigonometric functions, we know that \( \sin(60^{\circ}) = \sqrt{3}/2 \)

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions, often abbreviated as "sin." It relates the angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. The sine function is periodic with a period of 360° (or 2π radians). This means that its values repeat every full rotation. Key characteristics of the sine function include:

• It starts at 0 for an angle of 0° and increases to a maximum value of 1 at 90°.
• It returns to 0 at 180° and reaches a minimum value of -1 at 270°.
• At 360°, it completes its cycle and begins again.

For our problem, the sine of 60° is particularly important. By memorizing or using a unit circle, you know that:

• \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\)

This value can often be seen in various trigonometric expressions and exercises.

Cosine Function

The cosine function, abbreviated as "cos," is another fundamental trigonometric function. Like sine, cosine links the angle of a right triangle to a side length ratio: the adjacent side over the hypotenuse. It has a similar periodic nature, repeating every 360°.

• The cosine value is 1 at 0° and decreases to 0 at 90°.
• It further decreases to -1 at 180° before returning to 0 at 270°.
• By 360°, the cosine value is back at 1, completing its cycle.

In our exercise, we focus on the cosines of 60° and 120°:

• \(\cos(60^{\circ}) = \frac{1}{2}\)
• \(\cos(120^{\circ}) = -\frac{1}{2}\)

These values help simplify expressions involving cosine.

Angle Subtraction Formula

The angle subtraction formula is a powerful tool in trigonometry that allows us to simplify expressions involving trigonometric functions. It is particularly useful when we need to find the sine or cosine of an angle that is expressed as a difference between two angles.

• The angle subtraction formula for sine is:\(\sin(a - b) = \sin a \cos b - \cos a \sin b\)
• This formula helps break down complex expressions by transforming them into simpler ones, as shown in our exercise.

In the given example, the identity helped convert the entire expression into a simpler form, \(\sin(120^{\circ} - 60^{\circ})\)Which simplifies further to \(\sin(60^{\circ})\).Using this formula is an easy way to solve trigonometric equations and find exact values without relying on calculators. It also illustrates the elegance and symmetry inherent in trigonometry.