Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \csc 45^{\circ} $$

Step-by-Step Solution

Answer

\( \csc 45^{\circ} = \sqrt{2} \).

Step 1: Understanding Trigonometric Definitions

The cosecant function, \( \csc \theta \), is the reciprocal of the sine function. So, \( \csc \theta = \frac{1}{\sin \theta} \). Thus, to find \( \csc 45^{\circ} \), we need to first find \( \sin 45^{\circ} \).

Step 2: Evaluating Sine of 45 Degrees

We know from trigonometric ratios for special angles that \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \). This is a standard value that you should memorize.

Step 3: Calculating the Cosecant of 45 Degrees

Since \( \csc 45^{\circ} = \frac{1}{\sin 45^{\circ}} \), we substitute the value for \( \sin 45^{\circ} \). Therefore, \( \csc 45^{\circ} = \frac{1}{\frac{\sqrt{2}}{2}} \).

Step 4: Simplifying the Expression

To simplify \( \frac{1}{\frac{\sqrt{2}}{2}} \), we can multiply by the reciprocal to get \( \csc 45^{\circ} = \frac{2}{\sqrt{2}} \). We can further rationalize the denominator by multiplying both numerator and denominator by \( \sqrt{2} \), resulting in \( \frac{2\sqrt{2}}{2} \), which simplifies to \( \sqrt{2} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable involved. They are powerful tools in simplifying complex trigonometric expressions and solving trigonometric equations. Some of the fundamental trigonometric identities include:

• Pythagorean identities, such as \(\sin^2 \theta + \cos^2 \theta = 1\).
• Reciprocal identities, like \(\csc \theta = \frac{1}{\sin \theta}\).
• Quotient identities, for example, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

These identities allow us to express one trigonometric function in terms of others, which is very useful, especially when dealing with the problem of finding the exact value of trigonometric functions. By using these identities, we can often simplify expressions to make them easier to work with. For example, when solving \( \csc 45^\circ \), we use the reciprocal identity to transform it into a form involving \( \sin 45^\circ \). This makes finding the solution straightforward once we know \( \sin 45^\circ \).

Special Angles

Special angles are specific angles for which we know the exact values of the trigonometric functions without needing a calculator. These angles include \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\). Each of these angles is significant in trigonometry, leading to what we call 'unit circle' values.For example:

• \(\sin 45^\circ = \frac{\sqrt{2}}{2}\)
• \(\cos 45^\circ = \frac{\sqrt{2}}{2}\)
• \(\tan 45^\circ = 1\)

These values are derived from geometric principles and are consistent across all applications of trigonometry. Memorizing these values helps in quickly solving problems, such as determining \( \csc 45^\circ \), where knowing \( \sin 45^\circ \) directly leads us to the solution. By knowing these special angles and their trigonometric ratios, students can perform many calculations more efficiently.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions provide a way to represent trigonometric functions in terms of their reciprocal values. The main reciprocal trigonometric functions are cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each is defined as the reciprocal of a commonly used trigonometric function:

• Cosecant is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\)
• Secant is the reciprocal of cosine: \(\sec \theta = \frac{1}{\cos \theta}\)
• Cotangent is the reciprocal of tangent: \(\cot \theta = \frac{1}{\tan \theta}\)

These reciprocal functions have critical roles in various trigonometric calculations, especially when simplifying expressions or solving equations. In the exercise, finding \( \csc 45^\circ \) involved using the reciprocal identity \( \csc \theta = \frac{1}{\sin \theta} \). This identity allows us to transform a function into a simpler or more useful form, making the problem more approachable. Recognizing and using reciprocal functions can significantly enhance the breadth of problems a student can tackle successfully in trigonometry.