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The Pythagorean theorem is a fundamental principle in geometry, particularly useful in dealing with right-angled triangles. This theorem states that for any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. It can be expressed as: \[ a^2 + b^2 = c^2 \]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. This simple but powerful principle allows us to calculate the unknown side of a triangle if the other two are known.
In the context of trigonometric identities, this theorem underpins relationships like \[ \sin^2 \theta + \cos^2 \theta = 1 \]For angle \( B \) in our problem, using the fact that \( \cos B = \frac{3}{8} \), we applied this identity to find \( \sin B \). By rearranging the identity, \( \sin^2 B = 1 - \cos^2 B \), we were able to determine \( \sin B = \frac{\sqrt{55}}{8} \). Therefore, the Pythagorean theorem not only applies to side lengths but also extends to trigonometric functions through these identities.

Reciprocal identities are a key concept in trigonometry that relate the basic trigonometric functions to each other. They essentially provide a way to find the reciprocal of these functions easily:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities are enormously helpful when one function's value is known, allowing easy computation of others. In solving our problem, given \( \cos B = \frac{3}{8} \), we found \( \sec B \) using the reciprocal identity \( \sec B = \frac{1}{\cos B} = \frac{8}{3} \).
Similarly, once \( \sin B \) was determined, we applied its reciprocal identity to find \( \csc B = \frac{8}{\sqrt{55}} \). Rationalizing the denominator, it simplified to \( \frac{8\sqrt{55}}{55} \). The same process was used to calculate \( \cot B = \frac{3\sqrt{55}}{55} \) using the reciprocal of \( \tan B \). These identities simplify the process of finding one trigonometric function from another.

A right-angled triangle is a type of triangle that includes a 90-degree angle. It has three sides: the hypotenuse, opposite, and adjacent sides. The hypotenuse is the longest side opposite the right angle, while the other two are identified relative to a particular acute angle.

• Adjacent side: The side next to the angle considered, excluding the hypotenuse.
• Opposite side: The side opposite the angle considered.

In trigonometry, this triangle is vital because the basic trigonometric ratios are defined based on its side lengths:

• \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

These functions form the basis of trigonometric calculations. For angle \( B \) in our example, knowing \( \cos B \) directly gives us the ratio of the adjacent side to the hypotenuse, \( \frac{3}{8} \).
Use this understanding of a right-angled triangle to apply trigonometric functions and ultimately find the values needed in various trigonometric identities. Such knowledge can be extended to solve many geometric and real-world problems efficiently.