Problem 38 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \left(\sin 30^{\circ}\right)\left(\cos 60^{\circ}\right) $$

Step-by-Step Solution

Answer

The exact value is \(\frac{1}{4}\).

Step 1: Identify the Trigonometric Values

We need to recall the exact trigonometric values for the angles given. The sine of 30 degrees is known to be \(\sin 30^\circ = \frac{1}{2}\), and the cosine of 60 degrees is known to be \(\cos 60^\circ = \frac{1}{2}\).

Step 2: Set up the Expression Substituting Known Values

Substitute the known values into the expression \((\sin 30^\circ)(\cos 60^\circ)\). Thus, it becomes \(\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\).

Step 3: Simplify the Expression

To simplify the expression \(\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\), multiply the fractions: \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).

Key Concepts

Sine and Cosine Values

The values of sine (sin) and cosine (cos) are critical in trigonometry, especially when dealing with angles commonly encountered in problems, such as 30° and 60°. Understanding and memorizing these values can greatly simplify calculations:

• Sine of 30°: The sine function gives the vertical component (or opposite side) of an angle in a right triangle. For 30 degrees, \(\sin 30^{\circ} = \frac{1}{2}\).
• Cosine of 60°: Similarly, the cosine function gives the horizontal component (or adjacent side) of an angle. For 60 degrees, \(\cos 60^{\circ} = \frac{1}{2}\).

These values often appear in standardized tests and fundamental trigonometry problems, and recognizing them on sight can save time. Both sine and cosine are periodic functions, meaning they repeat their values at regular intervals. This repetition is key in analyzing patterns in trigonometric problems.

Angle Measurement

In trigonometry, angles are typically measured in degrees or radians. It's important to be comfortable transitioning between these two systems because certain scientific and mathematical fields prefer one over the other.

• Degrees: A full circle is 360 degrees, which is a common standard for everyday measurements like those on a protractor.
• Radians: In contrast, radians use the radius of the circle as the unit of measurement. One full circle is \(2\pi\) radians. Therefore, 30° is equivalent to \(\frac{\pi}{6}\) radians, and 60° is \(\frac{\pi}{3}\) radians.

Converting between degrees and radians is straightforward: \[\text{degrees} = \text{radians} \times \frac{180}{\pi}\]For most trigonometric calculations involving known values like 30° and 60°, working in degrees is more intuitive.

Trigonometric Multiplication

Trigonometric multiplication is the process of multiplying trigonometric values such as sine and cosine. When you multiply these values, as in the problem \(\sin 30^{\circ} \cdot \cos 60^{\circ}\), you deal mainly with fractions whether the values involve angles or any trigonometric expressions.

• Simple Multiplication: In the context of basic trigonometry, multiplying these fractions \(\frac{1}{2} \times \frac{1}{2}\) results in \(\frac{1}{4}\).
• Why Multiply?: Such multiplications appear in various trigonometric identities and simplify regular algebraic expressions, especially when resolving equations in more complex math problems.

Understanding how to deal with these products is crucial in finding values and simplifications in algebra and calculus, enabling more advanced mathematical explorations.