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Chapter 6: Problem 15

9鈥�32 Find the exact value of the trigonometric function. $$\csc \left(-630^{\circ}\right)$$

Short Answer

Expert verified
The exact value of \( \csc(-630^{\circ}) \) is 1.

Step by step solution

01

Understand the Problem

We need to find the exact value of the cosecant function, \( \csc \theta \), given the angle \( -630^{\circ} \).
02

Use Angle Reduction

To simplify the problem, we need to find an equivalent angle within the standard range \([0^{\circ}, 360^{\circ} )\). We do this by adding or subtracting multiples of \(360^{\circ}\) until the angle falls within this range. Compute \( -630^{\circ} + 720^{\circ} = 90^{\circ} \).
03

Recall the Cosecant Function

The cosecant of an angle \( \theta \) is the reciprocal of the sine of that angle. Thus, \( \csc \theta = \frac{1}{\sin \theta} \).
04

Find the Sine of the Angle

Now that we have the equivalent angle \( 90^{\circ} \), find \( \sin 90^{\circ} \). We know \( \sin 90^{\circ} = 1 \).
05

Calculate the Cosecant Value

Using the reciprocal identity, calculate \( \csc 90^{\circ} = \frac{1}{\sin 90^{\circ}} = \frac{1}{1} = 1 \).

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Key Concepts

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

Understanding Cosecant
The cosecant function, abbreviated as \( \csc \), is one of the six main trigonometric functions. It is defined as the reciprocal of the sine function. This means that for any given angle \(\theta\), \(\csc \theta = \frac{1}{\sin \theta}\). Understanding this relationship is crucial for solving problems involving cosecant.
  • Cosecant does not exist for angles where the sine value is zero. This is because dividing by zero is undefined in mathematics.
  • Common angles to consider are those where the sine value is either 1 or -1, such as \(90^{\circ}\) or \(270^{\circ}\).
Angle Reduction Simplified
Angle reduction is a technique used to simplify the computation of trigonometric functions for angles that are outside the standard range of \([0^{\circ}, 360^{\circ})\). By adjusting the angle by adding or subtracting full circles \((360^{\circ})\), the process brings the angle into a more manageable range.
  • This method is especially useful in periodic functions like sine and cosine, as their values repeat every \(360^{\circ}\).
  • For instance, with \(-630^{\circ}\), adding \(720^{\circ}\) (two full circles) results in \(90^{\circ}\).
Angle reduction simplifies computation and helps when visualizing the position of angles on the unit circle.
The Sine Function Explored
The sine function is fundamental to trigonometry and is represented as \( \sin \theta \). It outputs the y-coordinate of a point on the unit circle at a given angle \(\theta\) from the positive x-axis.
  • On the unit circle, \( \sin \theta \) reaches its maximum value of 1 at \(90^{\circ}\) and its minimum of -1 at \(270^{\circ}\).
  • The sine function is periodic with a period of \(360^{\circ}\), meaning it repeats its values every full circle.
  • Understanding that \( \sin 90^{\circ} = 1 \) is critical, as it allows us to immediately know \( \csc 90^{\circ} \) without further calculation.
Recognizing these features enables seamless computation of trigonometric values in various mathematical scenarios.

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Most popular questions from this chapter

Number of Solutions in the Ambiguous Case We have seen that when using the Law of Sines to solve a triangle in the SSA case, there may be two, one, or no solution(s). Sketch triangles like those in Figure 6 to verify the criteria in the table for the number of solutions if you are given \(\angle A\) and sides \(a\) and \(b\) . $$ \begin{array}{c|c}{\text { Criterion }} & {\text { Number of Solutions }} \\\ \hline a \geq b & {1} \\ {b>a>b \sin A} & {2} \\ {a=b \sin A} & {1} \\ {a

Height of a Rocket A rocket fired straight up is tracked by an observer on the ground a mile away. (a) Show that when the angle of elevation is \(\theta,\) the height of the rocket in feet is \(h=5280 \tan \theta\) (b) Complete the table to find the height of the rocket at the given angles of elevation. $$\begin{array}{|c|c|c|c|c|c|}\hline \theta & {20^{\circ}} & {60^{\circ}} & {80^{\circ}} & {85^{\circ}} \\ \hline h & {} & {} & {} \\\ \hline\end{array}$$

Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has latitude \(35^{\circ} \mathrm{N}\) and New Orleans, \(30^{\circ} \mathrm{N}\) . Find the distance between these two cities. (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Use the first Pythagorean identity to prove the second. [Hint: Divide by \(\cos ^{2} \theta . ]\)

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation \(\theta\) of a rainbow is always the same. It can be shown that \(\theta=4 \beta-2 \alpha\) where $$\sin \alpha=k \sin \beta$$ and \(\alpha=59.4^{\circ}\) and \(k=1.33\) is the index of refraction of water. Use the given information to find the angle of elevation \(\theta\) of a rainbow. (For a mathematical explanation of rainbows see Calculus, 5 th Edition, by James Stewart, pages \(288-289 .\) )
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