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Magazine Chapter 1: Problem 13
\(\csc 60^{\circ}\)
Short Answer
Expert verified
\( \csc 60^{\circ} = \frac{2\sqrt{3}}{3} \).
Step by step solution
01
Understanding the Function
The exercise asks for the cosecant of 60 degrees. The cosecant function, denoted as \( \csc \), is the reciprocal of the sine function. Therefore, \( \csc \theta = \frac{1}{\sin \theta} \). In this case, we need to find \( \sin 60^{\circ} \) first in order to compute \( \csc 60^{\circ} \).
02
Finding Sine of 60 Degrees
Recall that \( \sin \theta \) for 60 degrees is a standard trigonometric value. We have \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \). It's helpful to remember this common value from the unit circle or from an equilateral triangle split into two 30-60-90 triangles.
03
Calculating the Cosecant
Given \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), we use the reciprocal relationship to find \( \csc 60^{\circ} \). Thus, \( \csc 60^{\circ} = \frac{1}{\sin 60^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \).
04
Rationalizing the Denominator
The expression \( \frac{2}{\sqrt{3}} \) can be further simplified by rationalizing the denominator. To do this, multiply the numerator and the denominator by \( \sqrt{3} \): \( \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \). This is the simplified value of \( \csc 60^{\circ} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Values
Trigonometric values are essential in understanding angles and triangles. They relate the angles of a triangle to the lengths of its sides. In trigonometry, \(\sin\), \(\cos\), and \(\tan\) are the primary functions you will encounter. Each of these functions provides a ratio of two sides of a right-angled triangle.
Common angles you will frequently use are 30掳, 45掳, and 60掳, each corresponding to specific trigonometric values. These angles often appear in problems because their trigonometric values can be derived from special triangles or the unit circle.
The sine of 60掳, which is \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), is one such value. Recognizing these standard values can speed up calculations and provide a deeper understanding of trigonometric functions.
Common angles you will frequently use are 30掳, 45掳, and 60掳, each corresponding to specific trigonometric values. These angles often appear in problems because their trigonometric values can be derived from special triangles or the unit circle.
The sine of 60掳, which is \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), is one such value. Recognizing these standard values can speed up calculations and provide a deeper understanding of trigonometric functions.
Reciprocal Functions
Reciprocal functions in trigonometry are fascinating because they offer alternative perspectives on the main trigonometric functions. The primary reciprocal trigonometric functions are cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). These functions are defined as the reciprocals of sine, cosine, and tangent respectively.
For example:
For example:
- \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cot \theta = \frac{1}{\tan \theta}\)
Unit Circle
The unit circle is a powerful tool in trigonometry, giving insight into angle measures and trigonometric function values. It's a circle with a radius of one centered at the origin of a coordinate plane. The unit circle helps visualize how trigonometric functions repeat or mimic periodic cycles.
On the unit circle, the coordinates of any point (x, y) correspond to \(\cos \theta\) and \(\sin \theta\) respectively, for a given angle \(\theta\). This property makes the unit circle a quick reference for determining trigonometric values.
For \(\sin 60^\circ\), the y-value of the point on the unit circle is \(\frac{\sqrt{3}}{2}\). The unit circle ensures that these values are standardized and easy to remember, making calculations with trigonometric functions more straightforward.
On the unit circle, the coordinates of any point (x, y) correspond to \(\cos \theta\) and \(\sin \theta\) respectively, for a given angle \(\theta\). This property makes the unit circle a quick reference for determining trigonometric values.
For \(\sin 60^\circ\), the y-value of the point on the unit circle is \(\frac{\sqrt{3}}{2}\). The unit circle ensures that these values are standardized and easy to remember, making calculations with trigonometric functions more straightforward.
Rationalizing Denominators
Rationalizing denominators is a technique used to simplify expressions involving roots in the denominator. This practice is rooted in making calculations easier and results cleaner by eliminating irrational numbers from the denominator.
The process involves multiplying the numerator and denominator by the same root present in the denominator. In our case, for \(\frac{2}{\sqrt{3}}\), we multiply both parts by \(\sqrt{3}\), yielding \(\frac{2\sqrt{3}}{3}\).
Rationalizing keeps expressions neat and ready for further mathematical operations and applications. It reflects a fundamental step in making sure trigonometric solutions are presented in their most understandable form.
The process involves multiplying the numerator and denominator by the same root present in the denominator. In our case, for \(\frac{2}{\sqrt{3}}\), we multiply both parts by \(\sqrt{3}\), yielding \(\frac{2\sqrt{3}}{3}\).
Rationalizing keeps expressions neat and ready for further mathematical operations and applications. It reflects a fundamental step in making sure trigonometric solutions are presented in their most understandable form.
