Problem 28 - Precalculus with Calculus Previews

Question

If \(\csc x=\sec x,\) find all values of \(\tan x, \cot x, \sin\) \(x,\) and \(\cos x\).

Step-by-Step Solution

Answer

For \( x = \frac{\pi}{4} + n\pi \), \( \tan x = 1 \), \( \cot x = 1 \), \( \sin x = \frac{\sqrt{2}}{2} \), \( \cos x = \frac{\sqrt{2}}{2} \).

Step 1: Understanding the Given Equation

The problem states that \( \csc x = \sec x \). Recall the definitions of cosecant and secant: \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). So, the equation becomes \( \frac{1}{\sin x} = \frac{1}{\cos x} \).

Step 2: Cross-Multiplying

To eliminate the fractions, cross-multiply the equation \( \frac{1}{\sin x} = \frac{1}{\cos x} \) to obtain \( \sin x = \cos x \).

Step 3: Finding Angles Where Sine Equals Cosine

The equation \( \sin x = \cos x \) implies \( \tan x = 1 \), because \( \tan x = \frac{\sin x}{\cos x} \). Angles where this equality holds are \( x = \frac{\pi}{4} + n\pi \), where \( n \) is an integer.

Step 4: Calculating \( \sin x \) and \( \cos x \)

For \( x = \frac{\pi}{4} \) or any angle of the form \( x = \frac{\pi}{4} + n\pi \), \( \sin x = \cos x = \frac{\sqrt{2}}{2} \).

Step 5: Calculating \( \tan x \) and \( \cot x \)

Since \( \sin x = \cos x \), \( \tan x = \frac{\sin x}{\cos x} = 1 \) and \( \cot x = \frac{\cos x}{\sin x} = 1 \).

Key Concepts

Sine and Cosine

The sine and cosine functions lie at the heart of trigonometry. They are defined for a unit circle, where the radius of the circle is one unit. Each angle in trigonometry relates to a point on this circle, and sine and cosine are the coordinates of this point:

• The sine of an angle is the y-coordinate.
• The cosine of an angle is the x-coordinate.

When \[ \sin x = \cos x \]these two numbers are equal for specific angles. Such an equation holds true because it describes a scenario where the angle is \[\frac{\pi}{4} = 45^\circ\] or any angle congruent to it, considering all rotations around the circle (i.e., added multiples of \[\pi\]). At \[x = \frac{\pi}{4}\], both sine and cosine yield the value \[\frac{\sqrt{2}}{2}\]. This symmetry occurs because at \[45^\circ\], the segment or line in the unit circle makes equal projections onto both the x and y axes.Understanding this is crucial because it translates directly into equal ratios for sine and cosine, building a foundation for other trigonometric functions.

Secant and Cosecant

Secant and cosecant are the reciprocal functions of cosine and sine respectively. This means:

• Secant (\(\sec x\)) is defined as \(\frac{1}{\cos x}\).
• Cosecant (\(\csc x\)) is defined as \(\frac{1}{\sin x}\).

In our problem,\(\csc x = \sec x\)implies that both sine and cosine must take on reciprocal values which are equal at certain angles. This is due to the cross-multiplication method, where,\[\frac{1}{\sin x} = \frac{1}{\cos x} \]leads to the simplification,\[\sin x = \cos x.\]Thus, secant and cosecant illustrate how trigonometric identities can transform a problem into solvable parts. Simplifying equations using these identities can reveal angles and values where certain properties hold, like equal projections for sine and cosine.

Tangent and Cotangent

Tangent and cotangent are ratios derived from sine and cosine which capture important geometric properties:

• The tangent of an angle (\(\tan x\)) is the ratio \(\frac{\sin x}{\cos x}\).
• The cotangent of an angle (\(\cot x\)) is the inverse: \(\frac{\cos x}{\sin x}\).

When \(\sin x = \cos x\), both \(\tan x\)and \(\cot x\) are equal to 1. This is because their definitions simplify directly to 1, as the division of equals is always 1,\[\tan x = \frac{\sin x}{\cos x} = 1, \quad\cot x = \frac{\cos x}{\sin x} = 1.\]Thus, understanding tangent and cotangent is crucial to analyzing how angles and their complements balance through trigonometry. Tangent especially finds numerous applications as it represents slope and angle steepness in many physical and graphical scenarios.