Problem 48 - Algebra 2

Question

Solve each equation. $$ \operatorname{Arccos} \frac{\sqrt{2}}{2}=x $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{\pi}{4} \).

Step 1: Understand the Meaning of Arccos

The function \( \operatorname{Arccos} \) is the inverse of the cosine function. It returns the angle \( x \) in the range \( [0, \pi] \) whose cosine is equal to the given value.

Step 2: Relate the Value to a Known Angle

We know that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \). Thus, since arccos only takes values between \( 0 \) and \( \pi \), we identify \( \operatorname{Arccos} \frac{\sqrt{2}}{2} = \frac{\pi}{4} \).

Step 3: Verify the Solution

To verify, recall that \( \operatorname{Arccos}(y) = x \) implies \( \cos(x) = y \). Since \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), our solution \( x = \frac{\pi}{4} \) is verified as correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, linking angles in various geometric shapes, especially circles, with dimensions of those shapes. Primary trigonometric functions include sine, cosine, and tangent, and these form relationships between the angles and sides of right-angled triangles. Each function has its specific role and characteristics.

• Sine focuses on the ratio of the length of the opposite side to the hypotenuse.
• Cosine is the ratio of the adjacent side to the hypotenuse.
• Tangent takes the ratio of the opposite side to the adjacent longer side.

These functions are not limited to simple right-angled triangles; they extend into describing wave patterns and periodic rhythms observed in physics and engineering. Angles in trigonometric functions can be in degrees or radians, with radians being more prevalent in calculus and analytical work. Understanding these functions lays the groundwork for tackling more complex concepts, such as inverse trigonometric functions, allowing you to unlock angles based on specific ratios.

Cosine Function

Cosine, as part of the core trigonometric functions, plays a pivotal role in understanding angles and sides in geometry. The cosine function, denoted as \( \cos \), outputs a ratio based on a given angle. This ratio relates the length of the adjacent side of a right-angled triangle to the hypotenuse.

• Important properties include periodicity (repeating every \( 2\pi \)), symmetry, and its ability to model oscillations.
• It reaches a maximum value of 1 when the angle is 0 radians, and a minimum value of -1 at \( \pi \) radians.
• Common angles, like \( 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \), have well-known cosine values, crucial for solving problems quickly and accurately.

Through practice, recognizing the cosine function's role allows for efficient problem-solving in geometry and trigonometry. Knowing that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) is vital for understanding exercises involving inverse functions, as this provides a common reference point for identifying angles based on cosine values.

Inverse Functions

Inverse trigonometric functions are essential for determining angles from given ratios. These functions reverse the process of standard trigonometric functions. Specifically, the inverse of the cosine function is called arccosine, denoted as \( \operatorname{Arccos} \).

• \( \operatorname{Arccos}(x) \) yields an angle \( \theta \), meaning \( \cos(\theta) = x \).
• The commonly used range for the arccosine function is from \( 0 \) to \( \pi \), ensuring each input corresponds to a unique output and making it a true function.
• This range avoids confusion that would arise from cosine's periodic nature, which can otherwise repeat values.

In exercises like the given problem, identifying known angles from inverse values, such as \( \operatorname{Arccos} \frac{\sqrt{2}}{2} \) resulting in \( \frac{\pi}{4} \), demonstrates how inverse functions allow us to find specific angles with confidence. Mastery of inverse functions enables students to solve trigonometric equations and tackle complex mathematical challenges involving angle determinations.