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Chapter 5: Problem 14

Use the composite argument properties to show that the given equation is an identity. $$\sec \left(\theta-90^{\circ}\right)=\csc \theta$$ (Be clever!)

Short Answer

Expert verified
\( \sec(\theta - 90^\circ) = \csc(\theta) \) is an identity due to the co-function identity \( \cos(\theta - 90^\circ) = \sin(\theta) \).

Step by step solution

01

Understand the Composite Argument for Trigonometric Functions

The composite argument property for trigonometric functions describes how the function values change when the input angle is altered by adding or subtracting a common angle, in this case, 90 degrees. The secant function is related to the cosine, with the relationship being \( \sec(\theta) = \frac{1}{\cos(\theta)} \), and the cosecant function is related to the sine function, with the relationship \( \csc(\theta) = \frac{1}{\sin(\theta)} \) .
02

Apply Co-function Identities

Co-function identities state that \( \sin(\theta) = \cos(90^\circ - \theta) \) and \( \cos(\theta) = \sin(90^\circ - \theta) \) . Use these identities to relate the secant function and the cosecant function, noting that \( \sec(\theta - 90^\circ) = \frac{1}{\cos(\theta - 90^\circ)} \) and replace \( \cos(\theta - 90^\circ) \) with the co-function identity \( \sin(\theta) \).
03

Rewrite \(\sec(\theta - 90^\circ)\) using Co-function Identity

Rewrite the secant function as \( \sec(\theta - 90^\circ) = \frac{1}{\cos(\theta - 90^\circ)} = \frac{1}{\sin(\theta)} \), recognizing that \( \cos(\theta - 90^\circ) = \sin(\theta) \) from the co-function identity.
04

Show the Identity

Since \( \frac{1}{\sin(\theta)} \) is the definition of \( \csc(\theta) \) , we can conclude that \( \sec(\theta - 90^\circ) = \csc(\theta) \) , proving the given equation is indeed an identity.

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Key Concepts

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

Composite Argument Property
Understanding the composite argument property is essential for manipulating and simplifying trigonometric expressions. The composite argument property, also referred to as the angle addition and subtraction identities, guides us on how the trigonometric functions behave when we add or subtract an angle from the input.

For example, the cosine of a difference can be represented as: \[\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\]
This property becomes particularly handy when working with angles related to \(90^\circ\). Since \(90^\circ\) is a quarter turn on the unit circle, it shifts the function to its co-function, a key concept that we'll discuss next.
Co-function Identities
Co-function identities are a cornerstone of trigonometry, revealing an intrinsic symmetry in the trigonometric functions. These identities reflect a simple but profound relation: the sine of an angle is equal to the cosine of its complementary angle (the difference between \(90^\circ\) and the angle), and vice versa.

Expressed mathematically, for any angle \(\theta\): \[\sin(\theta) = \cos(90^\circ - \theta)\] \[\cos(\theta) = \sin(90^\circ - \theta)\]
Recognizing and applying these identities can simplify many trigonometric problems and is especially useful when converting between the secant and cosecant functions.
Secant Function
The secant function is somewhat less common in basic trigonometry but is equally important. It serves as the reciprocal of the cosine function. This relationship becomes apparent when you understand that the secant function can be defined as the hypotenuse divided by the adjacent side in a right triangle, which, in fact, is the inverse ratio of the cosine.

The secant function is written as: \[\sec(\theta) = \frac{1}{\cos(\theta)}\]
When the angle is shifted by \(90^\circ\), the secant of this angle is equivalent to another function, which brings us to the close relationship between the secant function and the next topic, the cosecant function.
Cosecant Function
The cosecant function is the complementary counterpart to the secant function, as it is the reciprocal of the sine function. This function can be visualized as the ratio of the length of the hypotenuse over the opposite side in a right-angled triangle. The cosecant is expressed mathematically as: \[\csc(\theta) = \frac{1}{\sin(\theta)}\]
In exercises involving co-function identities, recognizing the relationship between the secant and the cosecant functions can quickly simplify expressions, particularly when dealing with composite arguments that involve angle shifts, such as the \(90^\circ\) phase shift highlighted in the original exercise.

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Most popular questions from this chapter

Musical Note Problem: The Nett sisters, Cora and Clara, are in a band. Each one is playing the note \(A\). Their fricnd Tom is standing at a place where the notes arrive exactly a quarter cycle out of phase. If \(x\) is time in seconds, the function equations of Cora's and Clara's notes are Cora: \(y=100 \cos 440 \pi x\) Clara: \(y=150 \sin 440 \pi x\) PICTURE CANT COPY a. The sound Tom hears is the sum of Cora's and Clara's sound waves. Write an equation for this sound as a single cosine with a phase displacement. b. The amplitudes 100 and 150 measure the loudness of the two notes Cora and Clara are playing. Is this statement true or false? "Tom hears a note 250 units loud, the sum of 100 and \(150 .\) " Explain how you reached your answer. c. The frequency of the A being played by Cora and Clara is 220 cycles per second. Explain how you can figure this out from the two equations. Is the following true or false? "The note Tom hears also has a frequency of 220 cycles per second."

Sine Double Argument Property Derivation Problem: Starting with \(\sin 2 x=\sin (x+x)\) derive the property \(\sin 2 x=2 \sin x \cos x\)

Write the linear combination of cosine and sine as a single cosine with a phase displacement. $$y=\cos \theta-\sin \theta$$

Illustrate by numerical example that the double argument property is true by making a table of values. $$\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}$$

Use the composite argument properties with exact values of functions of special angles (such as \(30^{\circ}, 45^{\circ}, 60^{\circ}\) ) to show that these numerical expressions are exact values of \(\sin 15^{\circ}\) and \(\cos 15^{\circ} .\) Confirm numerically that the values are correct. $$\cos 15^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}$$
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