Problem 28 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cos \left(\tan ^{-1} \sqrt{3}\right)\)

Step-by-Step Solution

Answer

The value is 0.50.

Step 1: Understand the Inverse Tangent Function

We need to determine the angle \( \theta \) such that \( \tan(\theta) = \sqrt{3} \). Since the tangent of an angle gives the ratio of the opposite side to the adjacent side in a right triangle, \( \theta \) must be an angle where tan equals \( \sqrt{3} \).

Step 2: Determine the Angle in Radians

Recall standard angles and their tangent values. We know that \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \). Therefore, \( \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).

Step 3: Find the Cosine of the Angle

Now, calculate \( \cos\left(\frac{\pi}{3}\right) \). The cosine function gives the ratio of the adjacent side to the hypotenuse. We know that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

Step 4: Round to the Nearest Hundredth

The computed cosine value is \( 0.5 \), which is already rounded to the nearest hundredth.

Key Concepts

Angle Measurement in Radians

When measuring angles, we often use radians instead of degrees. Radians offer a natural and mathematical way to express angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

Imagine cutting a piece of string exactly the same length as a circle's radius and laying it along the outside edge of the circle. The angle subtended by this piece of string at the circle's center is 1 radian.

Converting between degrees and radians is straightforward. Since a full circle is 360 degrees and also equivalent to \(2\pi\) radians, we have this conversion:

• Degrees to Radians: Multiply by \(\frac{\pi}{180}\)
• Radians to Degrees: Multiply by \(\frac{180}{\pi}\)

Use radians for trigonometric computations, as they simplify calculus and other mathematical fields.

Cosine Function

The cosine function is one of the principal trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse.

For instance, in a right triangle with an angle \(\theta\), if the length of the adjacent side is 5 and the hypotenuse is 10, the cosine of \(\theta\) is \(\frac{5}{10} = 0.5\). This demonstrates how the function is calculated and provides insight into its practical uses.

Cosine values can range from -1 to 1 because they represent ratios of side lengths. It plays a critical role in understanding wave patterns, oscillations, and even alternating current electrical circuits. In our exercise, we calculated \(\cos\left(\frac{\pi}{3}\right)\), which equals 0.5, showing that the adjacent side is half the length of the hypotenuse for this angle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They simplify complex expressions and solve equations that involve trigonometric functions. Knowing these identities is essential for efficiently handling and understanding trigonometric problems.

Some common identities include:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle Sum Identity for Cosine: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• Double Angle Identity for Cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \)

Each of these identities helps to transform and relate various trigonometric expressions and can help solve for unknown angles or simplify equations. In our exercise, understanding \(\tan^{-1}\) and how it relates to angles allows us to substitute and find cosine values, integrating these identities to better solve complex problems.