Problem 25 - Algebra 2

Question

Find the exact value of each function. $$ \frac{\cos 60^{\circ}+\sin 30^{\circ}}{4} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Trigonometric Values

Recognize the standard angle values: \( \cos 60^{\circ} \) and \( \sin 30^{\circ} \). For \( \cos 60^{\circ} \), the value is \( \frac{1}{2} \). For \( \sin 30^{\circ} \), the value is also \( \frac{1}{2} \).

Step 2: Substitute Values into the Expression

Substitute the known values into the expression. This gives \( \frac{\frac{1}{2} + \frac{1}{2}}{4} \).

Step 3: Simplify the Numerator

Add the two fractions in the numerator: \( \frac{1}{2} + \frac{1}{2} = 1 \).

Step 4: Divide by the Denominator

Divide the result obtained in Step 3 by 4: \( \frac{1}{4} \).

Key Concepts

Cosine

The cosine function is one of the fundamental trigonometric functions, frequently used in various mathematical applications. Cosine measures the ratio of the adjacent side to the hypotenuse in a right triangle. This function is essential to understanding the relationships between angles and side lengths in triangles.
In the exercise above, the angle given is 60 degrees. The cosine of an angle, represented as \( \cos \theta \), is a specific value that is often memorized or easily accessible in trigonometric tables. For \( \cos 60^{\circ} \), the value is \( \frac{1}{2} \). This means that for an angle of 60 degrees in a right triangle, the length of the adjacent side is half of the hypotenuse.

• Related terms you might encounter are: cosine rule, adjacent side, and hypotenuse.
• Cosine values are important in understanding wave patterns, oscillations, and alternating current circuits.

Always keep in mind that cosine values can range from -1 to 1.

Sine

The sine function is another crucial trigonometric function, alongside cosine. In a right triangle, sine is the ratio of the length of the opposite side to the hypotenuse. Understanding sine helps in solving problems involving angles and distances.
In the provided problem, we look at the sine of a 30-degree angle, noted as \( \sin 30^{\circ} \). This value simplifies to \( \frac{1}{2} \), meaning that the opposite side is half the length of the hypotenuse for a 30-degree angle. Just like cosine, sine values are often memorized or tabulated for quick reference in mathematical computations.

• Sine is useful in various fields such as physics for describing waves, sound, and light.
• Sine and cosine functions are often used together to resolve vectors and forces.

It's important to acknowledge that sine values also range from -1 to 1, defining the north and south positions on a unit circle.

Angle Values

Finding accurate trigonometric values often involves knowing specific angle values. Angles like 30° and 60° are commonly used in trigonometry due to their occurrence in equilateral and right triangles. These angles have standard sine and cosine values that come from the special properties of the respective triangles.
Understanding these angle values is critical. In the step-by-step solution, the angle values were \(60^{\circ}\) and \(30^{\circ}\). The trigonometric functions then transform these angles into meaningful information in the form of ratios. For practical scenarios, these angle values help in analyzing periodicity, where the trigonometric function repeats itself, making computations efficient and straightforward.

• Recognize these angles in the context of the unit circle, where they correspond to specific points.
• These angles form the basis for many other calculations using trigonometric identities and formulas.

Knowing exact values for key angles allows for quick and accurate analysis of more complex mathematical problems.