Problem 54 - Algebra 2

Question

Find each value. $$ \cos \left(\cos ^{-1} \frac{4}{7}\right) $$

Step-by-Step Solution

Answer

\( \frac{4}{7} \)

Step 1: Understanding Inverse Cosine

The expression \( \cos^{-1} \frac{4}{7} \) is asking for the angle whose cosine value is \( \frac{4}{7} \). This angle is in the range of inverse cosine, which is usually between \(0\) and \(\pi\).

Step 2: Apply Cosine to Inverse Cosine

Once we have the angle given by \( \cos^{-1} \frac{4}{7} \), applying cosine to this angle should simply return \( \frac{4}{7} \), as we are essentially asking for the cosine of the angle which has a cosine value of \( \frac{4}{7} \).

Step 3: Final Simplification

Therefore, \( \cos \left( \cos^{-1} \frac{4}{7} \right) \) simplifies directly to \( \frac{4}{7} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are mathematical tools used to find angles when given a trigonometric ratio. Specifically, they help us work backward from the value to the angle itself. When dealing with inverse cosine, denoted as \( \cos^{-1} \), you are finding the angle whose cosine is a given value. For example, \( \cos^{-1} \frac{4}{7} \) means finding the angle \( \theta \) such that \( \cos(\theta) = \frac{4}{7} \).

These inverse functions are essential because they allow us to solve situations where the direction of calculation is reversed—moving from a ratio to an angle, rather than vice versa. Most inverse trigonometric functions are restricted to specific ranges because trigonometric functions are periodic and many values can repeat. For \( \cos^{-1} \), the range is typically between \( 0 \) and \( \pi \). Keeping the range restricted ensures that each value has one unique angle, making calculations consistent and predictable.

Cosine Function

The cosine function is one of the primary trigonometric functions, denoted as \( \cos \). It relates the ratio of the adjacent side to the hypotenuse in a right triangle to an angle. In the context of the unit circle, the cosine of an angle is the x-coordinate of a point on the circle.

The cosine function is periodic, meaning it repeats its values in a regular pattern over intervals. Specifically, \( \cos(\theta) \) has a period of \( 2\pi \), meaning it repeats every \( 360^\circ \) or \( 2\pi \) radians. This periodic nature allows the cosine function to model wave-like phenomena, such as light and sound waves.

• In the unit circle, \( \cos(\theta) \) equals the x-coordinate of the point corresponding to the angle \( \theta \).
• For angles between \( 0 \) and \( \pi \), \( \cos \) decreases from 1 to -1.

Understanding the cosine function is crucial for solving problems involving angles and their related sides.

Right Triangle Trigonometry

Right triangle trigonometry focuses on the relationships between the angles and sides of right triangles. One of the main applications is to resolve angles and sides using trigonometric functions like sine, cosine, and tangent. In a right triangle, each function relates a pair of sides to an angle.

• The cosine function relates the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• The sine function corresponds to the opposite side over the hypotenuse.
• Tangent uses the opposite side over the adjacent side.

These relationships help solve for unknown angles or sides, which is particularly helpful in various fields such as physics, engineering, and architecture.

By recognizing how these functions behave within a right triangle, one can easily compute missing measurements, perform coordinate transformations, and understand wave patterns. Right triangle trigonometry is foundational for further studies in both trigonometry and calculus.