Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-10, find the exact value without using a calculator. $$ \arccos \left(\frac{\sqrt{2}}{2}\right) $$

Step-by-Step Solution

Answer

\( \arccos \left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4} \).

Step 1: Identify the Range and Definition of arccos

The function \( \arccos(x) \) outputs an angle whose cosine is \( x \). The range of \( \arccos \) is \([0, \pi]\). This means we are searching for an angle in this range whose cosine is \( \frac{\sqrt{2}}{2} \).

Step 2: Recall Special Angles and Their Cosine Values

Recall from trigonometry that \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \). Since \( \frac{\pi}{4} \) is within the range \([0, \pi]\), it is a candidate for our solution.

Step 3: Verify the Angle

Verify that the angle \( \frac{\pi}{4} \) is indeed within the permissible range for the \( \arccos \) function and that its cosine matches the given value. Since \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), this confirms it as the solution.

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics that deals with the relationships between the angles and sides of triangles. It is fundamental in understanding how to navigate between radians and degrees, crucial for various scientific applications.
At its core, trigonometry involves basic functions like sine, cosine, and tangent. These functions help in calculating angles and distances in triangles, particularly right-angled triangles. They also extend to larger concepts, like wave functions and oscillations in physics.
In terms of **inverse trigonometric functions**, these are quite essential too. They allow us to find angles when given a trigonometric ratio. For example, the inverse cosine function, \( \arccos(x) \), helps find the angle whose cosine is \( x \). This is particularly useful when working on problems that involve reversing trigonometric functions to determine the original angle or measurement, as seen in the given exercise.

• **Sine function** - relates opposite to the hypotenuse.
• **Cosine function** - connects adjacent to the hypotenuse.
• **Tangent function** - links opposite to adjacent.

These core concepts are key to solving many kinds of mathematical problems, including those in engineering and physics.

Cosine Function

The cosine function (\( \cos \)) is a cornerstone of trigonometry. It relates the angle of a right triangle to the ratio of its adjacent side over its hypotenuse. This makes it incredibly useful in situations where you need to find one side of a triangle when given an angle and the other side.
For the cosine function, the range is typically between -1 and 1, as it corresponds to ratios of triangle sides. Graphically, the cosine function forms a wave-like pattern, repeating every \(2\pi\) radians. Understanding this can be useful when interpreting periodic behavior in both mathematics and applied sciences.
Inverse cosine, often noted as \( \arccos(x) \), is used to reverse the cosine function. It calculates the angle that would have a specific cosine value. The range of \( \arccos(x) \) is limited to \([0, \pi] \), making it particularly useful for determining specific angles in standard mathematical contexts. This is nicely demonstrated in the exercise where we find \( \arccos(\frac{\sqrt{2}}{2}) \).

• **Range of cosine**: -1 to 1.
• **Key angle values**: 0, \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), etc.
• **Applications**: Triangle solving, sound waves, etc.

The cosine function and its inverse provide powerful tools for both theoretical and practical calculations.

Special Angles

Special angles are specific angle measures that frequently appear in trigonometry, allowing us to derive exact values for trigonometric functions quickly and accurately. These angles, typically measured in radians, include \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \(\frac{\pi}{2}\), among others. Understanding these can greatly simplify computations without a calculator.
These angles are known for producing neat, rational values when used as inputs in the sine, cosine, or tangent functions. For example, the problem \( \arccos(\frac{\sqrt{2}}{2}) \) can be solved by recalling that this cosine value corresponds to the special angle \(\frac{\pi}{4}\). Knowing the cosine values of special angles helps in confirming the solution's validity.

• **\(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\)** - Common special angles.
• **Exact trigonometric values** - Memorize them for efficiency.
• **Direct application** - Useful in inverse trigonometry problems.

Being familiar with special angles enables students to handle trigonometric problems confidently, transforming complex calculations into simpler tasks.