Problem 27 - Algebra and Trigonometry

Question

Evaluate the expression without using a calculator. $$ \sin 30^{\circ} \cos 60^{\circ}+\sin 60^{\circ} \cos 30^{\circ} $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Recall Known Values

Let's first recall the trigonometric values for certain angles: - \( \sin 30^\circ = \frac{1}{2} \)- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)- \( \cos 60^\circ = \frac{1}{2} \)

Step 2: Substitute Trigonometric Values

Substitute these values into the expression \( \sin 30^\circ \cos 60^\circ + \sin 60^\circ \cos 30^\circ \):\[ \frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \].

Step 3: Simplify the Expression

Calculate each component of the expression:- \( \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \)- \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4} \)Now add these results together: \( \frac{1}{4} + \frac{3}{4} = \frac{4}{4} \).

Step 4: Present the Final Result

The resulting simplified expression is \( \frac{4}{4} = 1 \). This completes the evaluation of the given expression.

Key Concepts

Angle Values

Understanding angle values is key to solving trigonometric problems. In trigonometry, certain angles are so frequently used that their sine and cosine values are well known and remembered by many students. Common angles include 30°, 45°, and 60°.Knowing these values by heart helps in solving identities quickly. For example:

• For 30°, the sine value is \( \sin 30^\circ = \frac{1}{2} \) and the cosine value is \( \cos 30^\circ = \frac{\sqrt{3}}{2} \).
• For 60°, \( \sin 60^\circ = \frac{\sqrt{3}}{2} \) and \( \cos 60^\circ = \frac{1}{2} \).

These values are based on the unit circle and help in evaluating expressions like \( \sin 30^\circ \cos 60^\circ + \sin 60^\circ \cos 30^\circ \) without any calculator assistance. When learning trigonometry, it's beneficial to create a mental or even a physical cheat sheet to recall these values easily.

Trigonometric Functions

Trigonometric functions such as sine (\( \sin \)) and cosine (\( \cos \)) are fundamental in mathematics, especially geometry. They describe the relationship between the angles and sides of triangles, most specifically in a right triangle.- **Sine Function (\( \sin \))**: It relates the angle to the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.- **Cosine Function (\( \cos \))**: Similarly, it relates the angle to the ratio of the length of the adjacent side to the hypotenuse.These functions are critical for analyzing periodic behavior in mathematics. They, along with the tangent and their respective inverses, form the basis for understanding circular and oscillatory phenomena in physics and engineering. Their prominence in evaluating expressions like \( \sin 30^\circ \cos 60^\circ + \sin 60^\circ \cos 30^\circ \) illustrates their broad application in mathematical problem-solving.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics. It involves rewriting expressions in a simpler and often more useful form without changing their value. Let's break down our example:The expression to simplify is \( \sin 30^\circ \cos 60^\circ + \sin 60^\circ \cos 30^\circ \). After replacing the angle values, we compute:

• \( \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \)
• \( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4} \)

Once these products are calculated, adding them together yields \( \frac{1}{4} + \frac{3}{4} = \frac{4}{4} \), simplifying finally to 1.This process of simplifying ensures that mathematical expressions become more manageable. It’s a fundamental process for solving equations and functions efficiently, making it a cornerstone of mathematical education.