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The secant function, denoted as \( \sec \), is one of the six primary trigonometric functions. It is the reciprocal of the cosine function and is defined for angles where the cosine is not zero. To put it simply:
\[ \sec \alpha = \frac{1}{\cos \alpha} \]

This means that if you know the value of the secant function, you can easily determine the value of the cosine function by taking the reciprocal. For instance, in the given exercise, we know that \( \sec \alpha = \frac{10}{3} \).

To find \( \cos \alpha \), we simply take the reciprocal of \( \frac{10}{3} \):

\[ \cos \alpha = \frac{1}{\sec \alpha} = \frac{3}{10} \]

The cosine function, denoted as \( \cos \), is another primary trigonometric function and relates an angle of a right triangle to the lengths of the adjacent side and the hypotenuse.

The cosine of an angle \( \alpha \) in a right triangle is given by:

\[ \cos \alpha = \frac{ \text{adjacent side} }{ \text{hypotenuse} } \]

This function is fundamental in trigonometry and has various applications in physics, engineering, and other fields.

Notice how knowing the cosine enables us to understand other trigonometric functions like the secant, sine, and tangent through their relationships.

In the exercise, with \( \sec \alpha = \frac{10}{3} \), we've calculated \( \cos \alpha \) by finding its reciprocal, resulting in \( \cos \alpha = \frac{3}{10} \).

Reciprocal identities are crucial to understanding trigonometric functions and their relationships. Essentially, each primary trigonometric function has a reciprocal:

• The reciprocal of the sine function \( \sin \) is the cosecant function \( \csc \).
• The reciprocal of the cosine function \( \cos \) is the secant function \( \sec \)
• The reciprocal of the tangent function \( \tan \) is the cotangent function \( \cot \)

In formula terms:

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

By understanding these identities, solving trigonometric equations becomes more manageable. For example, if you know \( \sec \alpha \), you can find \( \cos \alpha \) quickly, as illustrated in the exercise.

Here, with \( \sec \alpha = \frac{10}{3} \), we used the reciprocal identity to find:

\[ \cos \alpha = \frac{1}{\sec \alpha} = \frac{3}{10} \]