Problem 42 - Algebra 2 and Trigonometry

Question

In \(39-50,\) find the smallest positive value of \(\theta\) to the nearest degree. $$ \cos \theta=0.7660 $$

Step-by-Step Solution

Answer

The smallest positive value of \( \theta \) is approximately 40 degrees.

Step 1: Understanding the Problem

We need to find the smallest positive angle \( \theta \) for which \( \cos \theta = 0.7660 \). This requires using the inverse cosine function to determine \( \theta \).

Step 2: Using the Inverse Cosine Function

To find \( \theta \), we use the inverse cosine (cosine inverse) function: \( \theta = \cos^{-1}(0.7660) \). Calculating this will give us the angle in degrees.

Step 3: Calculating the Angle

Using a calculator, compute \( \theta = \cos^{-1}(0.7660) \). This should give a result which you'll round to the nearest degree.

Step 4: Rounding the Result

Round the computed angle to the nearest whole number to find \( \theta \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics, particularly when solving problems related to angles and their measurements. They help us find the angle that corresponds to a specific trigonometric value, such as sine, cosine, or tangent.

For example, given the equation \( \cos \theta = 0.7660 \), we use the inverse cosine, also denoted as \( \cos^{-1} \), to find the angle \( \theta \). It effectively reverses the action of a cosine function, translating a given cosine value back into the angle that yields it.

These functions are extremely useful not only in pure mathematics but also in practical applications like physics and engineering, where determining precise angles is crucial. Remember, when using inverse functions, the range of the result is significant— for inverse cosine, the output is usually in the interval \([0, \pi]\) radians or \([0, 180]\) degrees.

Cosine Function

The cosine function is one of the primary trigonometric functions and is fundamentally linked to the geometry of right-angled triangles and the unit circle. In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

On the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle. This makes cosine a crucial part of representing periodic phenomena, like waves.

When computing an angle \( \theta \) for which \( \cos \theta = 0.7660 \), this suggests that the angle corresponds to a particular spot on the unit circle with a cosine value of 0.7660. Using a calculator, \( \cos^{-1}(0.7660) \) gives us the precise angle \( \theta \), which is beneficial in many calculations.

Angle Measurement

Angle measurement is a critical concept in mathematics and many fields that rely on precise geometrical data, such as engineering or construction. Angles can be measured in degrees or radians. Here, we are using degrees, where a full circle is 360 degrees.

To convert a decimal angle measurement to an integer, we round it to the nearest whole number. This is especially important when exactness is necessary for practical applications, like setting the angle of a machine part.

For instance, in the problem where \( \cos \theta = 0.7660 \), once we calculate \( \theta = \cos^{-1}(0.7660) \), we need to 'round the computed angle to the nearest degree.' This practice ensures accuracy and consistency in measurement, essential for real-world applications.