91Ó°ÊÓ

To find an acute angle given a trigonometric ratio is a fundamental application of trigonometry. An acute angle is an angle measuring less than 90 degrees. When dealing with a function like cosine, the task typically involves determining the angle whose ratio (cosine value) is known.
In our scenario, the cosine of angle \(B\) is 0.3090. To determine the acute angle that corresponds to this ratio, we use a calculator to find the inverse cosine, also known as arccosine. This inverse operation allows us to backtrack from the value of the cosine to the actual angle in degrees.

• Identify the trigonometric function and its given ratio. For angle \(B\), we have \(\cos B = 0.3090\).
• Use a scientific calculator to compute the inverse cosine of 0.3090.
• Ensure that your calculator is in degree mode to receive the correct angle measurement.

This simple procedure helps us solve for \(B\) where \(B\) is an acute angle, and the output will demonstrate a measurable degree less than 90.

In trigonometry, ratios such as sine, cosine, and tangent are associated with the angles of right triangles. These ratios establish a direct relationship between the angles and sides of these triangles. The cosine ratio specifically is the length of the adjacent side divided by the length of the hypotenuse.
Given the equation \(\cos B = 0.3090\):

• This tells us that for angle \(B\) in a right triangle, the length of the side adjacent to the angle is 0.309 times the length of the hypotenuse.
• Trigonometric functions are foundational for predicting and analyzing various geometric and physical phenomena.
• The inverse trigonometric functions allow us to calculate angles when we know certain side ratios.

Understanding these ratios helps us achieve a deeper grasp of angles and their practical applications, both in mathematics and in real-world problems.

Rounding angles is an important step in many calculations, especially when dealing with measurements where precision is less critical. For the problem at hand, after finding the angle using the inverse cosine, it is necessary to round to the nearest tenth of a degree.
The process of rounding involves:

• Identifying the tenth's place in the angle result, let's say it is \(x.456\) for an example value.
• Looking at the digit in the hundredths place (which is 5 in our example).
• Applying the rule of rounding: if the number in the hundredths place is 5 or greater, increase the tenth's place by one. Here, \(x.456\) becomes \(x.5\).

This practice allows for standardized results that are easier to compare and analyze across different results and exercises, providing clarity and consistent precision in calculations.