Problem 19 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \operatorname{Arccos} \frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

The angle is 0.52 radians.

Step 1: Understanding Arccos Function

The function \( \operatorname{Arccos}(x) \) gives us the angle whose cosine is \( x \). It's the inverse of the cosine function, defined in the range \([0, \pi]\) for real number inputs.

Step 2: Identify Cosine Value

We need to find an angle \( \theta \) such that \( \cos(\theta) = \frac{\sqrt{3}}{2} \). This is a common cosine value which corresponds to a well-known angle.

Step 3: Recall Special Angles

Given \( \cos(\theta) = \frac{\sqrt{3}}{2} \), the angle \( \theta \) is known to be \( \frac{\pi}{6} \) radians. In the defined range for Arccos, \( \theta = \frac{\pi}{6}\) is the angle we need as it's the only angle in the range \([0, \pi]\) that satisfies this condition.

Step 4: Conclude and Round

Since \( \frac{\pi}{6} = 0.5235987756 \) radians, rounding to the nearest hundredth gives \( 0.52 \). Therefore, \( \operatorname{Arccos}\left(\frac{\sqrt{3}}{2}\right) = 0.52 \) radians.

Key Concepts

Radian Measure

Radian measure is a way of describing angles based on the radius of a circle. In this system, angles are measured in terms of the circle's radius. One full circle is equal to \( 2\pi \) radians, which is approximately 6.283 radians. This differs from degrees, where a full circle is 360 degrees.
Understanding radians is crucial for working with trigonometric functions, as it provides a clear mathematical approach.

• 1 radian is the angle at the center of a circle that subtends an arc equal in length to the circle's radius.
• This unit simplifies many mathematical equations in physics and engineering.

When dealing with problems in trigonometry like the one here, it's important to provide answers in radians, especially in contexts where precision is necessary.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential for finding angles when the ratio of sides in a triangle is known. These functions include arcsine, arccosine, and arctangent.
The arccosine function, or \( \operatorname{Arccos}(x) \), specifically finds the angle whose cosine is the number \( x \). It is the inverse of the cosine function. Its output range is between \( 0 \) and \( \pi \) radians.

• Arccosine is used to solve equations where the cosine value is known, but the angle is not.
• It helps in transitioning from a function value back to its angle form.

Understanding the range restrictions is key, as it helps to avoid multiple possible solutions which would otherwise arise in other unwieldy ranges.

Special Angles

Special angles are key angles in trigonometry that have well-known sine, cosine, and tangent values. These angles commonly include \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \) radians, which correspond to \( 30\degree \), \( 45\degree \), and \( 60\degree \) respectively.
Understanding these angles is useful since they frequently appear in trigonometric exercises and problems.

• \( \frac{\pi}{6} \) radians (or 30°) has a cosine value of \( \frac{\sqrt{3}}{2} \).
• These angles simplify calculations since their trigonometric values often involve simple fractions or radicals.

Mastery of these special angles allows for quick problem-solving and recognition of patterns in trigonometric identities.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the ratios of its sides and include sine, cosine, and tangent. The cosine function, used here, measures the adjacent side over the hypotenuse in a right triangle.
Understanding these relationships forms the basis for more advanced mathematical applications and problem-solving.

• Cosine is periodic, with a period of \( 2\pi \), meaning it repeats every \( 360\degree \) or full rotation.
• The cosine of common angles yields values that are key in calculations across various problems.

These functions are not limited to triangles and are integral to various fields such as physics, engineering, and computer graphics for modeling wave patterns and cyclical phenomena.