Problem 66 - Precalculus : Building Concepts and Connections

Question

Find the measure of the acute angle \(\theta\) for which the sine or cosine is given. Let \(\beta\) be an acute angle with \(\cos \beta=b .\) Find \(\sec \beta\) and \(\sin \left(90^{\circ}-\beta\right)\) in terms of \(b\)

Step-by-Step Solution

Answer

The value of \(\sec\beta\) in terms of \(b\) is \(1/b\), and the value of \(\sin(90° - \(\beta\))\) in terms of \(b\) is \(b\).

Step 1: Understand the problem

Given \(\cos\beta = b\), which where \(b\) could be cosine of the acute angle or sine of the same. The task is to find \(\sec\beta\) and \(\sin\)(90° - \(\beta\)).

Step 2: Solve for \(\sec\beta\)

\(\sec\beta\) is defined as the reciprocal of \(\cos\beta\). as \(\cos\beta = b\), Therefore, \(\sec\beta = 1/b\).

Step 3: Solve for \(\sin(90° - \(\beta\))\)

\(\sin(90° - \(\beta\))\) can be interpreted as \(\cos\(\beta\)\). As we know \(\cos\beta = b\), Therefore, \(\sin(90° - \(\beta\)) = b\)

Step 4: Conclusion

The value of \(\sec\beta\) in terms of \(b\) is \(1/b\) and \(\sin(90° - \(\beta\))\) in terms of \(b\) is \(b\).

Key Concepts

Understanding Acute Angles

An acute angle is an angle that measures less than 90 degrees. In the context of trigonometry, these angles are particularly important because most initial trigonometric functions are defined in terms of angles less than 90 degrees.

Here are some key points to remember about acute angles:

• Acute angles always have positive trigonometric values.


• In right triangles, the non-right angles are acute.


• They are fundamental in trigonometric identities and transformations, including complementary angle pairs like \( \sin(90^\circ - \theta) = \cos\theta \).

Recognizing the role of acute angles helps in understanding how these angles relate to trigonometric functions.

Exploring the Cosine Function

The cosine function relates the angle of a right triangle with the ratio of the adjacent side to the hypotenuse. For an acute angle \( \beta \), \( \cos\beta \) helps you understand this relationship.

In mathematical terms:

• The formula is \( \cos\beta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).


• Cosine values range from 0 to 1 for acute angles.


• This function is also crucial in defining other trigonometric functions like secant, where \( \sec\beta = \frac{1}{\cos\beta} \).

Understanding cosine is essential as it is the foundation for complex trigonometric transformations and is used in various real-world applications, such as calculating angles of elevation and depression.

Decoding the Secant Function

The secant function is an essential trigonometric ratio, representing the reciprocal of the cosine function. For any given acute angle \( \beta \), the secant is expressed in relation to the cosine.

Key aspects of the secant function include:

• The formula is \( \sec\beta = \frac{1}{\cos\beta} \).


• As \( \cos\beta \) varies, \( \sec\beta \) either increases or decreases accordingly.


• Secant is especially valuable in solving trigonometric equations where reciprocals are involved.

By comprehending how secant relates to cosine, you can solve problems that involve reciprocal trigonometric identities and increase your understanding of the interplay between different trigonometric functions.