The Secant, Cosecant, and Cotangent Functions - AP PreCalculus Study Guide

The Core Idea: The Secant, Cosecant, and Cotangent Functions

The secant, cosecant, and cotangent functions are extensions of the primary trigonometric functions: sine, cosine, and tangent. The fundamental concept is that these three functions are defined as the reciprocals of the sine, cosine, and tangent functions. Rather than introducing entirely new principles, this topic establishes a set of definitional relationships that allow for a deeper analysis of trigonometric behavior.

The secant function, $sec(x)$, is defined as the reciprocal of the cosine function. The cosecant function, $csc(x)$, is the reciprocal of the sine function. Finally, the cotangent function, $cot(x)$, is the reciprocal of the tangent function. Because these functions are defined by division, their domains are inherently restricted. A reciprocal function is undefined wherever its corresponding base function is equal to zero. Understanding these definitions and the resulting domain restrictions is the central focus of this topic.

Key Formulas: Reciprocal and Quotient Identities

The definitions of the secant, cosecant, and cotangent functions are based on reciprocal and quotient relationships with the sine, cosine, and tangent functions. These definitions are the foundational formulas for this topic.

• The Secant Function: The secant of an angle $x$ is the reciprocal of the cosine of $x$.

  $$\sec (x)=\frac{1}{\cos (x)}$$

• The Cosecant Function: The cosecant of an angle $x$ is the reciprocal of the sine of $x$.

  $$\csc (x)=\frac{1}{\sin (x)}$$

• The Cotangent Function: The cotangent of an angle $x$ is the reciprocal of the tangent of $x$.

  $$\cot (x)=\frac{1}{\tan (x)}$$

  Because the tangent function is defined as $tan(x)=sin(x)/cos(x)$, the cotangent function can also be expressed as the quotient of the cosine and sine functions.

  $$\cot (x)=\frac{\cos (x)}{\sin (x)}$$

Understanding Domain Restrictions

A critical aspect of the secant, cosecant, and cotangent functions is that their domains are not all real numbers. Because these functions are defined as reciprocals, they are undefined whenever their denominator is equal to zero. This leads to specific exclusions from their domains.

• Domain of the Secant Function: The secant function is defined as $sec(x)=1/cos(x)$. Therefore, the function is undefined for any value of $x$ where $cos(x)=0$. The cosine function is zero at $x=\pi /2$, $x=3\pi /2$, and so on. The general form for these values is $x=\pi /2+k\pi$, where $k$ is any integer. The domain of $sec(x)$ is all real numbers $x$ such that $cos(x)\ne 0$.

• Domain of the Cosecant Function: The cosecant function is defined as $csc(x)=1/sin(x)$. Therefore, the function is undefined for any value of $x$ where $sin(x)=0$. The sine function is zero at $x=0$, $x=\pi$, $x=2\pi$, and so on. The general form for these values is $x=k\pi$, where $k$ is any integer. The domain of $csc(x)$ is all real numbers $x$ such that $sin(x)\ne 0$.

• Domain of the Cotangent Function: The cotangent function can be defined as $cot(x)=cos(x)/sin(x)$. Similar to the cosecant function, its denominator is $sin(x)$. Therefore, the cotangent function is also undefined for any value of $x$ where $sin(x)=0$. The domain of $cot(x)$ is all real numbers $x$ such that $sin(x)\ne 0$.

Core Concepts & Rules

• Secant is the Reciprocal of Cosine: The function $sec(x)$ is fundamentally defined by the relationship $sec(x)=1/cos(x)$.

• Cosecant is the Reciprocal of Sine: The function $csc(x)$ is fundamentally defined by the relationship $csc(x)=1/sin(x)$.

• Cotangent is the Reciprocal of Tangent: The function $cot(x)$ is defined as $cot(x)=1/tan(x)$.

• Cotangent as a Quotient: An equivalent and often more useful definition for cotangent is the ratio $cot(x)=cos(x)/sin(x)$.

• Secant Domain Rule: The domain of $sec(x)$ excludes all values of $x$ for which $cos(x)=0$.

• Cosecant and Cotangent Domain Rule: The domains of $csc(x)$ and $cot(x)$ exclude all values of $x$ for which $sin(x)=0$.

Step-by-Step Example 1: Evaluating Reciprocal Functions

Problem: Given that for a certain angle $\theta$, $sin(\theta )=-12/13$ and $cos(\theta )=5/13$, find the values of $csc(\theta )$, $sec(\theta )$, and $cot(\theta )$.

Step 1: Calculate $csc(\theta )$

• Recall the definition of the cosecant function: $csc(\theta )=1/sin(\theta )$.

• Substitute the given value of $sin(\theta )$ into the formula.

  $$\csc (\theta )=\frac{1}{-12/13}$$

• To divide by a fraction, multiply by its reciprocal.

  $$\csc (\theta )=1\cdot \left(-\frac{13}{12}\right)=-\frac{13}{12}$$

Step 2: Calculate $sec(\theta )$

• Recall the definition of the secant function: $sec(\theta )=1/cos(\theta )$.

• Substitute the given value of $cos(\theta )$ into the formula.

  $$\sec (\theta )=\frac{1}{5/13}$$

• Multiply by the reciprocal.

  $$\sec (\theta )=1\cdot \left(\frac{13}{5}\right)=\frac{13}{5}$$

Step 3: Calculate $cot(\theta )$

• Recall the quotient definition of the cotangent function: $cot(\theta )=cos(\theta )/sin(\theta )$.

• Substitute the given values of $cos(\theta )$ and $sin(\theta )$.

  $$\cot (\theta )=\frac{5/13}{-12/13}$$

• To divide these fractions, multiply the numerator by the reciprocal of the denominator.

  $$\cot (\theta )=\frac{5}{13}\cdot \left(-\frac{13}{12}\right)=-\frac{5\cdot 13}{13\cdot 12}=-\frac{5}{12}$$

Solution:

$csc(\theta )=-13/12$, $sec(\theta )=13/5$, and $cot(\theta )=-5/12$.

Step-by-Step Example 2: Determining the Domain

Problem: Determine the domain of the function $f(x)=sec(x)$ on the interval $[0,2\pi ]$. Express the domain using interval notation.

Step 1: State the Definition and Identify the Restriction

• The secant function is defined as $f(x)=sec(x)=1/cos(x)$.

• The function is undefined when its denominator is zero. Therefore, we must find all values of $x$ in the interval $[0,2\pi ]$ for which $cos(x)=0$.

Step 2: Solve for the Values Where the Function is Undefined

• We need to solve the equation $cos(x)=0$ for $x$ in the interval $[0,2\pi ]$.

• The values of $x$ for which the cosine is zero are $x=\pi /2$ and $x=3\pi /2$. Both of these values are within the specified interval $[0,2\pi ]$.

Step 3: Exclude the Restricted Values and Write the Domain

• The domain of $f(x)=sec(x)$ on $[0,2\pi ]$ is all numbers in this interval except for $\pi /2$ and $3\pi /2$.

• To express this in interval notation, we "remove" these points, creating open intervals around them.

• The domain starts at $0$ (inclusive) and goes up to $\pi /2$ (exclusive).

• It then continues from $\pi /2$ (exclusive) to $3\pi /2$ (exclusive).

• Finally, it continues from $3\pi /2$ (exclusive) to $2\pi$ (inclusive).

• Combining these gives the final domain in interval notation.

Solution:

The domain of $f(x)=sec(x)$ on the interval $[0,2\pi ]$ is $[0,\pi /2)U(\pi /2,3\pi /2)U(3\pi /2,2\pi ]$.

Using Your Calculator

Graphing calculators typically do not have dedicated buttons for $sec$, $csc$, and $cot$. To evaluate or graph these functions, you must use their reciprocal definitions.

To evaluate $sec(x)$:

• Use the relationship $sec(x)=1/cos(x)$.

• For example, to calculate $sec(0.5)$, you would type 1 / cos(0.5) into your calculator. Make sure your calculator is in the correct mode (radians or degrees).

To evaluate $csc(x)$:

• Use the relationship $csc(x)=1/sin(x)$.

• For example, to calculate $csc(\pi /4)$, you would type 1 / sin(π/4).

To evaluate $cot(x)$:

• Use either $cot(x)=1/tan(x)$ or $cot(x)=cos(x)/sin(x)$. The second form is often more stable as it avoids the vertical asymptotes of the tangent function itself.

• For example, to calculate $cot(30°)$, you could type `1 / tan(30°)or $cos(30°) / sin(30°).

When graphing these functions, you enter the reciprocal form into the Y= editor. For instance, to graph $y = csc(x), you would enter Y1 = 1/sin(x)`. The calculator will correctly render the graph, including the vertical asymptotes where $sin(x)=0$.

AP Exam Quick Hit

Common Question Types

• Direct Reciprocal Calculation: You will be given the value of a sine or cosine function and asked for the value of its corresponding reciprocal function.

  • Example: If $sin(x)=-0.8$, what is the value of $csc(x)$? (Answer: $1/-0.8=-1.25$)

• Identifying Domain Restrictions: You will be asked to find the values for which a reciprocal trigonometric function is undefined, or to state the domain of the function over a given interval.

  • Example: For which of the following values of $x$ is $cot(x)$ undefined? (A) $\pi /2$ (B) $3\pi /4$ (C) $\pi$ (D) $5\pi /4$. (Answer: (C), because $sin(\pi )=0$)

• Simplifying Expressions: You may be asked to simplify an expression containing both primary and reciprocal trigonometric functions by applying the definitions.

  • Example: Simplify the expression $tan(x)\ast cos(x)\ast csc(x)$. (Answer: $(sin(x)/cos(x))\ast cos(x)\ast (1/sin(x))=1$, for values where the expression is defined).

Common Mistakes

• Confusing Secant and Cosecant: A very common error is to incorrectly pair the reciprocal functions, for instance, assuming $sec(x)$ is the reciprocal of $sin(x)$. Remember: the "co-" functions do not pair up. $sec(x)$ pairs with $cos(x)$, and $csc(x)$ pairs with $sin(x)$.

• Mistaking Reciprocal for Inverse: Students often confuse $csc(x)$ with the inverse sine function, $si{n}^{-1}(x)$ (or $arcsin(x)$). The notation $si{n}^{-1}(x)$ means "the angle whose sine is x," whereas $csc(x)$ means $1/sin(x)$. These are completely different concepts.

• Ignoring Domain Restrictions: A frequent mistake is to perform a calculation without first checking if the function is defined for the given input. For example, stating that $cot(0)=1/tan(0)=1/0$ is incorrect; the function is simply undefined at $x=0$.

• Calculator Input Errors: When using a calculator, typing $co{s}^{-1}(x)$ (the inverse cosine button) when you intend to calculate $sec(x)$. You must always input 1 / cos(x) for the secant function.