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Chapter 1: Problem 57

(a) \(\csc \theta=\frac{2 \sqrt{3}}{3}\) (b) \(\sin \theta=\frac{\sqrt{2}}{2}\)

Short Answer

Expert verified
\[ \theta = \frac{\pi}{3}, \frac{2\pi}{3} \] in the case of \( \csc \theta \) and \( \theta = \frac{\pi}{4}, \frac{3\pi}{4} \) in the case of \( \sin \theta \)

Step by step solution

01

Solving for \( \csc \theta \)

Since \( \csc \theta \) is the reciprocal of \( \sin \theta \), we can write an equivalent equation with \( \sin \theta = \frac{1}{\csc \theta} \). Substituting the given value, we get \( \sin \theta = \frac{3}{2 \sqrt{3}} \). Simplifying the right hand side, we can rationalize the denominator by multiplying by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get \( \sin \theta = \frac{\sqrt{3}}{2} \)
02

Finding the angle for \( \sin \theta = \frac{\sqrt{3}}{2} \)

Looking at the unit circle, we know there are two angles in the interval \( 0 \leq \theta < 2\pi \) for which \( \sin \theta = \frac{\sqrt{3}}{2} \). These are \( \theta = \frac{\pi}{3} \) (or 60 degrees) and \( \theta = \frac{2\pi}{3} \) (or 120 degrees)
03

Solving for \( \sin \theta = \frac{\sqrt{2}}{2} \)

Looking at the unit circle again, there are two angles in the interval \( 0 \leq \theta < 2\pi \) for which \( \sin \theta = \frac{\sqrt{2}}{2} \). These are \( \theta = \frac{\pi}{4} \) (or 45 degrees) and \( \theta = \frac{3\pi}{4} \) (or 135 degrees)

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Key Concepts

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

Cosecant Function
The cosecant function, denoted as \( \csc \theta \), is one of the six primary trigonometric functions. It is the reciprocal of the sine function: \( \csc \theta = \frac{1}{\sin \theta} \). This means that for any angle \( \theta \), you can find its cosecant by taking the reciprocal of its sine value.

This function is most useful when dealing with triangles, specifically right triangles. It helps in finding the length of the hypotenuse in relation to the opposite side. Remember, the cosecant function is undefined whenever the sine value is 0, as division by zero is undefined in mathematics.
  • Formula: \( \csc \theta = \frac{1}{\sin \theta} \)
  • Key feature: Reciprocal of sine
  • Use: Relates hypotenuse with the opposite side
Sine Function
The sine function, represented as \( \sin \theta \), is one of the basic foundational elements in trigonometry. It relates an angle in a right triangle to the length of the opposite side and the hypotenuse.

Sine is essential for understanding concepts in circles and periodic functions. The function ranges between -1 and 1, repeating every \( 2\pi \) radians, or 360 degrees, known as its period. In simpler terms, it describes how an angle's height (opposite side in a triangle) changes as the angle increases or decreases.
  • Formula: \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
  • Range: -1 to 1
  • Period: \( 2\pi \) radians
  • Applications: Waves, circles, oscillations
Unit Circle
The unit circle is a vital concept in trigonometry with a radius of 1, centered at the origin of a coordinate plane. It is used to define trigonometric functions for all real numbers as well as to visualize angles and their corresponding values for sine and cosine.

On the unit circle, the angle \( \theta \) is measured from the positive x-axis, and the coordinates of the point where the terminal side of the angle intersects the circle provide cosine and sine values.
  • Radius: 1 unit
  • Coordinates: \((\cos \theta, \sin \theta)\)
  • Angles: Measured from positive x-axis
  • Purpose: Visualize and compute trigonometric functions
Angle Measurement
Angle measurement is fundamental for analyzing and utilizing trigonometric functions. Angles are typically measured in degrees or radians, with one full rotation around a circle being 360 degrees or \( 2\pi \) radians.

Radians offer a mathematical advantage since they relate directly to the radius of the circle. It鈥檚 important to understand both systems as some problems are simpler in degrees, and others, especially involving calculus and analysis, benefit from radians.
  • Degree: 360 for a full circle
  • Radian: \( 2\pi \) for a full circle
  • Conversion: \( 180^\circ = \pi \) radians
  • Utility: Facilitates calculations in geometry and trigonometry

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Most popular questions from this chapter

Use a graphing utility to graph the function. $$ f(x)=-\arcsin 2 x $$

In Exercises 19-34, use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arccos (-0.41) $$

In Exercises 59-68, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) $$ \tan \left(\arccos \frac{x}{3}\right) $$

$$ \text { In Exercises 49-58, find the exact value of the expression. } $$ $$ \sin \left(\arctan \frac{3}{4}\right) $$

In Exercises 19-34, use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin 0.45 $$
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