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Chapter 1: Problem 17

\(\sec 45^{\circ}\)

Short Answer

Expert verified
\(\sec 45^{\circ} = \sqrt{2}\)

Step by step solution

01

Understanding the Function

The cosecant function, also known as secant, is the reciprocal of the cosine function. To find \( \sec 45^{\circ} \), we first need the value of \( \cos 45^{\circ} \).
02

Finding \(\cos 45^{\circ}\)

To find \(\cos 45^{\circ}\), we refer to the unit circle or standard trigonometric values. We know that \(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\).
03

Reciprocal to Find \(\sec 45^{\circ}\)

Now that we know \(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\), we take the reciprocal to find \(\sec 45^{\circ}\). This gives us \(\sec 45^{\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions are essential mathematical functions that relate the angles of a triangle to the lengths of its sides. They are usually defined for right-angled triangles and include sine, cosine, tangent, as well as their reciprocals: cosecant, secant, and cotangent. These functions help us understand the relationships between angles and side lengths.
In a right triangle,
  • Sine ( \( \sin \theta \) ): The ratio of the opposite side to the hypotenuse.
  • Cosine ( \( \cos \theta \) ): The ratio of the adjacent side to the hypotenuse.
  • Tangent ( \( \tan \theta \) ): The ratio of the opposite side to the adjacent side.
  • Secant ( \( \sec \theta \) ): The reciprocal of cosine, or \( \frac{1}{\cos \theta} \).
Understanding these basic functions and their properties is crucial since they create a strong foundation for more complex trigonometric principles. They play a significant role in various fields like physics, engineering, and astronomy.
Unit Circle
The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin (0,0) on the coordinate plane. This simple yet powerful tool helps us visualize and understand trigonometric functions across different angles. When dealing with the unit circle,
  • The x-coordinate of a point on the unit circle corresponds to the cosine of the angle.
  • The y-coordinate corresponds to the sine of the angle.
  • It allows us to define trigonometric functions for all real numbers, not just within a triangle.
For example, when finding \( \cos 45^{\circ} \) using the unit circle, we look at the coordinates corresponding to that angle and find that \( \cos 45^{\circ} = \frac{1}{\sqrt{2}} \). This value is particularly useful when we need to find secant, as secant is simply the reciprocal of cosine.
Reciprocal Identities
Reciprocal identities are important in trigonometry, providing relationships between the main trigonometric functions and their reciprocals. These identities help simplify calculations and support the understanding of trigonometric behaviour.
  • Secant as a Reciprocal: Since secant is the reciprocal of cosine, \( \sec \theta = \frac{1}{\cos \theta} \). This identity is used when calculating \( \sec 45^{\circ} \). Knowing \( \cos 45^{\circ} = \frac{1}{\sqrt{2}} \), the \( \sec 45^{\circ} = \sqrt{2} \).
  • Other Reciprocals: The other primary reciprocal identities are cosecant, \( \csc \theta = \frac{1}{\sin \theta} \), and cotangent, \( \cot \theta = \frac{1}{\tan \theta} \).
Remembering these identities is key to solving trigonometric problems effectively, ensuring that steps like finding the reciprocal are straightforward.

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