Problem 15 - Algebra and Trigonometry

Question

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. $$ \sin 18^{\circ} \cos 27^{\circ}+\cos 18^{\circ} \sin 27^{\circ} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{2}}{2}\)

Step 1: Recognize the Formula

The given expression \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \) can be recognized as the sine addition formula. Recall the sine addition formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \).

Step 2: Identify the Angles

In the expression \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \), we identify \( A = 18^{\circ} \) and \( B = 27^{\circ} \), so \( A + B = 45^{\circ} \).

Step 3: Apply the Formula

Using the sine addition formula from Step 1, we substitute the angles identified in Step 2: \( \sin(18^{\circ} + 27^{\circ}) = \sin 45^{\circ} \).

Step 4: Find the Exact Value

We know from trigonometric values that \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

Key Concepts

Sine Addition Formula

The sine addition formula is a vital tool in trigonometry. It helps us express the sine of a sum of angles in a simplified form. This formula is written as: \[ \sin(A + B) = \sin A \cos B + \cos A \sin B. \] This means if you have two angles, \( A \) and \( B \), the sine of their sum can be rewritten into a combination of sines and cosines of the individual angles.

• \( \sin A \) is the sine of the first angle.
• \( \cos B \) is the cosine of the second angle.

In our exercise, you see this at work: the expression \( \sin 18^{\circ} \cos 27^{\circ} + \cos 18^{\circ} \sin 27^{\circ} \) mirrors the format of the sine addition formula. This allows us to substitute and simplify it to \( \sin(18^{\circ} + 27^{\circ}) \), helping us find the sine of a combined angle effortlessly.

Exact Trigonometric Values

Finding the exact trigonometric values means determining numeric expressions for sine, cosine, or tangent of commonly used angles. These values are frequently involved in calculations in trigonometry, and understanding them reduces computation time. Take the angle \( 45^{\circ} \) as an example; it has a well-known exact sine value which is \( \frac{\sqrt{2}}{2} \). This is because \( 45^{\circ} \) is derived from common geometric principles, often associated with an isosceles right triangle.

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• This is valid for both sine and cosine due to the symmetry of the angles.

Recognizing these values makes problem solving much quicker. In our exercise, once the expression is simplified to \( \sin 45^{\circ} \), you instantly know its exact value based on memorized math fundamentals of special angles.

Angle Addition

Angle addition involves combining two angles into one. With trigonometry, it allows us to utilize identities to simplify expressions. For instance, in an exercise like ours, angle addition facilitated the simplification of the expression from multiple products of trig functions into just one: a sine function evaluated at a single angle. Here's the process broken down:

• Identify \( A \) and \( B \) in the context, here \( 18^{\circ} \) and \( 27^{\circ} \), respectively.
• Add these angles: \( A + B = 45^{\circ} \).
• Use the sine addition identity to evaluate \( \sin 45^{\circ} \).

Angle addition is helpful in simplifying complex trigonometric expressions. It also aids in understanding the relationships between different trigonometric values and angles through identities like the sine addition. This not only streamlines calculations, but also deepens comprehension of how angles contribute to overall trigonometric functions.