Problem 60 - Precalculus : Building Concepts and Connections

Question

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. $$\cos \theta=\frac{\sqrt{3}}{2}$$

Step-by-Step Solution

Answer

The acute angle \(\theta\) for which \(\cos \theta=\frac{\sqrt{3}}{2}\) is 30 degrees.

Step 1: Recall the Unit Circle

Recall that the cosine of an angle in the unit circle corresponds to the x-coordinate of a point on the unit circle. Standard angles should be readily recalled, or use a reference such as a trigonometric table or unit circle chart.

Step 2: Match Value to Standard Angle

Find the standard angle(s) whose cosine is \(\frac{\sqrt{3}}{2}\). Cosine is positive in the first and fourth quadrants, but we only consider the first quadrant as we are looking for an acute angle. Cos(30 degrees) equals \(\frac{\sqrt{3}}{2}\).

Step 3: State The Solution

The acute angle \(\theta\) therefore is 30 degrees, as cos(30 degrees)=\(\frac{\sqrt{3}}{2}\).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry used to define sine, cosine, and tangent for all real numbers. Imagine a circle on a coordinate plane with a radius of one unit. The center of this circle is at the origin, where the x-axis and y-axis intersect (0,0). Because the radius is one, every point on the circle has coordinates that correspond to particular trigonometric sine and cosine values.

• In the unit circle, the cosine of an angle is represented by the x-coordinate of the corresponding point on the circle.
• The sine of the angle is represented by the y-coordinate.

These coordinates help interpret how angles correspond to specific coordinates on the circle, which can directly translate into determining values of \( \sin(\theta) \) and \( \cos(\theta) \). Understanding the unit circle allows us to quickly identify trigonometric ratios of standard angles, such as 30°, 45°, and 60°, which are frequently encountered in trigonometry.

Delving into Cosine

Cosine is one of the primary trigonometric functions, symbolized as \( \cos(\theta) \). It is used to determine the adjacent side of a right triangle with respect to the angle in question. On the unit circle, as discussed previously, the value of cosine corresponds to the x-coordinate.

A few key points about cosine include:

• In the first quadrant, cosine values are positive. This is where acute angles, angles less than 90°, are generally found.
• Cosine values vary between -1 and 1.

For our exercise, knowing that \( \cos(\theta) = \frac{\sqrt{3}}{2} \) indicates a standard angle. Recognizing these standard values is important as they often appear in calculations involving the unit circle. For instance, cosines of 30° and 60° are \( \frac{\sqrt{3}}{2} \) and \( \frac{1}{2} \) respectively, reflecting their x-coordinates on the unit circle.

Identifying Acute Angles

Acute angles are crucial in many areas of mathematics and geometry, and they are angles less than 90 degrees. They are often encountered in right triangles and many practical scenarios, including the layout plans of buildings and in the angular assessment of physical structures.

Understanding a few key aspects can aid in grasping acute angles:

• In the context of the unit circle and trigonometric functions, acute angles are typically found in the first quadrant where both sine and cosine values are positive.
• Common acute angles in trigonometry include 30°, 45°, and 60°.

In our exercise, knowing a trigonometric function value like \( \cos(\theta) \) helps pinpoint the exact angle. Recognizing typical cosine or sine values allows students to match them to the right acute angle, as in the provided example, where \( \cos(30°) = \frac{\sqrt{3}}{2} \). Knowledge of these standard angles is invaluable for quick reference and solving trigonometric problems efficiently.