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Chapter 6: Problem 99

Show that \(\csc \left(\frac{\pi}{2}+\theta+2 n \pi\right)=\sec \theta, n\) an integer.

Short Answer

Expert verified
The identity holds due to co-function identities and angle periodicity: \( \csc\left(\frac{\pi}{2} + \theta + 2n\pi\right) = \sec(\theta) \).

Step by step solution

01

Apply the Co-Function Identity

Recall that the cosecant function can be expressed in terms of secant using the co-function identity for angles. The formula is \[ \csc\left(\frac{\pi}{2} + \theta\right) = \sec(\theta) \]. Since adding any multiple of \(2\pi\) to an angle corresponds to a complete rotation and does not change the angle's trigonometric values, the inequality holds for any integer \(n\).
02

Simplify the Angle

Consider the function \( \csc\left(\frac{\pi}{2} + \theta + 2n\pi \right) \). By periodicity of the sine function, \( \sin(\alpha + k\cdot2\pi) = \sin(\alpha) \) for any integer \(k\). Thus \( \csc\left(\frac{\pi}{2} + \theta + 2n\pi \right) = \csc\left(\frac{\pi}{2} + \theta \right) \).
03

Use the Co-Function Formula

We already know from Step 1 that \[ \csc\left(\frac{\pi}{2} + \theta\right) = \sec\theta \]. So, substituting back, \[ \csc\left(\frac{\pi}{2} + \theta + 2n\pi\right) = \sec(\theta) \].
04

Conclude the Proof

Since the transformations applied did not change the identity and are valid due to periodicity and co-function identities, we can conclude that \[ \csc\left(\frac{\pi}{2} + \theta + 2n\pi\right) = \sec(\theta) \] for any integer \(n\).

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Key Concepts

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

Co-Function Identities
Co-function identities are a crucial component of trigonometry. They relate the trigonometric functions of complementary angles.
  • Two angles are complementary if their sum is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.
  • For example, the sine of an angle is equal to the cosine of its complement, and vice versa.
  • The co-function identity for the cosecant function states that \( \csc\left(\frac{\pi}{2} + \theta\right) = \sec(\theta) \).
The identity highlights how a shift involving a right angle (\( \frac{\pi}{2} \)) can transform one trigonometric function into another. This principle is valuable in proofs involving trigonometric identities, like the one provided in the exercise.
Angle Periodicity
Angle periodicity refers to how certain angles in trigonometry repeat their values at regular intervals. This principle arises because trigonometric functions are periodic.
  • The sine, cosine, secant, and cosecant functions all have a period of \( 2\pi \) radians, or \( 360^\circ \).
  • This means these functions "reset" every full circle around the unit circle, repeating values after a \( 2\pi \) interval.
In the exercise, we use periodicity to simplify the expression \( \csc(\frac{\pi}{2} + \theta + 2n\pi) \). Since adding \( 2n\pi \) merely involves full rotations, it does not affect the value, enabling simplification directly to \( \csc(\frac{\pi}{2} + \theta) \). This concept ensures that the function's properties remain consistent despite multiple rotations.
Trigonometric Functions
Trigonometric functions are fundamental in studying angles and their relations. They are used in various mathematical applications including geometry, calculus, and physics.
  • Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)).
  • These functions can be represented on the unit circle, which is a circle with a radius of one centered at the origin on a coordinate plane.
In this exercise, specifically, the cosecant and secant functions play a central role. The goal was to demonstrate that the function \( \csc(\frac{\pi}{2} + \theta + 2n\pi) \) simplifies to \( \sec(\theta) \), showing the intertwined nature of trigonometric identities. Understanding these relationships is key to mastering trigonometry.

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Most popular questions from this chapter

Solve each trigonometric equation on \(0^{\circ} \leq \theta<360^{\circ} .\) Express solutions in degrees and round to two decimal places. $$\cos (2 x)+\frac{1}{2} \sin x=0$$

In calculus, one technique used to solve differential equations consists of the separation of variables. For example, consider the equation \(x^{2}+3 y \frac{f(y)}{g(x)}=0,\) which is equivalent to \(3 y f(y)=-x^{2} g(x) .\) Here each side of the equation contains only one type of variable, either \(x\) or \(y\) Use the sum and difference identities to separate the variables in each equation. $$\cos (x-y)=0$$

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Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$2 \sin ^{2} x+3 \cos x=0$$
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