Problem 82 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the first section of the next chapter. The exercises use identities, introduced in Section 5.2, that enable you to rewrite trigonometric expressions so that they contain only sines and cosines: $$ \begin{aligned} \csc x &=\frac{1}{\sin x} & & \sec x=\frac{1}{\cos x} \\ \tan x &=\frac{\sin x}{\cos x} & & \cot x=\frac{\cos x}{\sin x} \end{aligned} $$ Rewrite each expression by changing to sines and cosines. Then simplify the resulting expression. $$ \sec x \cot x $$

Step-by-Step Solution

Answer

The given expression \(\sec x \cot x\) simplifies to \(\csc x\).

Step 1: Express the secant function in terms of cosines

Use the identity \[ \sec x=\frac{1}{\cos x} \] to replace \(\sec x\) in the expression. This gives us \[ \sec x \cot x = \frac{1}{\cos x} \cot x \].

Step 2: Express the cotangent function in terms of cosines and sines

Now replace the cotangent function \(\cot x\) using the identity \[ \cot x=\frac{\cos x}{\sin x} \]. This replacement in our equation gives us \[ \frac{1}{\cos x} \cdot \frac{\cos x}{\sin x} \].

Step 3: Simplification

Simplify the equation by multiplying through. The \(\cos x\) on the numerator and denominator cancel out, leaving \[ \frac{1}{\sin x} \]. This is in fact the definition of the function cosecant, so the expression simplifies to \[ \frac{1}{\sin x} \] or equivalently \(\csc x\).

Key Concepts

Sine and Cosine

Trigonometric functions like sine and cosine are the foundation of trigonometry, relating angles to sides of a right triangle.

The sine function, denoted as \(\sin x\), represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. The cosine function, \(\cos x\), on the other hand, gives the ratio of the length of the adjacent side to the hypotenuse.

In the unit circle, sine and cosine also define coordinates of a point where the terminal side of an angle intersects the circle. Here, \(\sin x\) corresponds to the y-coordinate and \(\cos x\) to the x-coordinate. This circle-based definition extends their application from just right-angled triangles to any angle, even beyond 90 degrees.

Secant and Cosecant

While not as commonly used as sine and cosine, the secant \(\sec x\) and cosecant \(\csc x\) functions are their reciprocal counterparts.

Secant is the reciprocal of cosine, whereas cosecant is the reciprocal of sine. Specifically, \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\). These relationships mean that for any non-zero sine or cosine value, secant and cosecant can represent an angle's relationship to the hypotenuse from a different perspective. Despite their less frequent appearance, secant and cosecant play vital roles in trigonometric identities and solving complex trigonometric equations.

Tangent and Cotangent

The tangent \(\tan x\) and cotangent \(\cot x\) functions provide ratios that are distinct from sine and cosine, offering alternative ways to understand angles and sides in triangles.

Tangent relates the opposite side to the adjacent side, defined as \(\tan x = \frac{\sin x}{\cos x}\).

Relationship Between Tangent and Cotangent

Cotangent is the reciprocal of tangent and also the ratio of the adjacent side to the opposite side, expressed as \(\cot x = \frac{\cos x}{\sin x}\). They are extremely useful for simplifying trigonometric expressions involving angles, as demonstrated by how \(\tan x\) and \(\cot x\) interact with each other and with \(\sin x\) and \(\cos x\) in advanced trigonometric problems.

Trigonometric Expressions Simplification

Simplifying trigonometric expressions is a fundamental skill in mathematics, requiring an understanding of how to manipulate and combine different trigonometric functions.

The objective is often to rewrite expressions in terms of basic functions like sine and cosine or to reduce an expression to its simplest form. As in our example with \(\sec x \cot x\), by applying known identities, we effectively simplified the expression to \(\csc x\), which is much easier to work with.

Strategies for Simplification

Key strategies include using fundamental identities, seeking common denominators, and canceling out terms when possible. The process increases the clarity and often the computational tractability of trigonometric problems. Simplification also enables the comparison and integration of trigonometric functions into broader mathematical contexts.