📐 Level 1 - Topic 7: Reciprocal Trigonometric Functions (Cosecant, Secant, Cotangent) 🔄

1) Beyond Sine, Cosine, Tangent - Introducing Reciprocals 🔄

We've learned about the three primary trigonometric functions: sine, cosine, and tangent. But in trigonometry, there are also three related functions called the reciprocal trigonometric functions. These are cosecant, secant, and cotangent. They are, as the name suggests, simply the reciprocals of sine, cosine, and tangent, respectively.

The Reciprocal Trigonometric Functions are defined as follows:

Cosecant (csc): Reciprocal of sine (sin).
Secant (sec): Reciprocal of cosine (cos).
Cotangent (cot): Reciprocal of tangent (tan).

Why Reciprocal Functions?

You might wonder why we need these extra functions if they are just reciprocals. There are several reasons:

Completeness: It provides a complete set of six trigonometric ratios, covering all possible ratios that can be formed from the sides of a right triangle (and generalized to all angles).
Mathematical Convenience: In certain formulas and contexts, using cosecant, secant, or cotangent can simplify expressions or make equations easier to work with.
Historical Reasons: Historically, all six trigonometric functions were recognized and used. While sine, cosine, and tangent are often considered primary today, the reciprocal functions are still important in many areas of mathematics and applications.

2) Defining Cosecant, Secant, and Cotangent - Formulas and Relationships 📐

Let's formally define cosecant, secant, and cotangent in terms of sine, cosine, tangent, and also in terms of opposite, adjacent, and hypotenuse sides in a right triangle (for acute angles).

Definitions and Formulas:

Cosecant of \( \theta \) (csc \( \theta \)):

\( \csc(\theta) = \frac{1}{\sin(\theta)} \)

In a right triangle: \( \csc(\theta) = \frac{\text{Hypotenuse}}{\text{Opposite}} \)
Secant of \( \theta \) (sec \( \theta \)):

\( \sec(\theta) = \frac{1}{\cos(\theta)} \)

In a right triangle: \( \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}} \)
Cotangent of \( \theta \) (cot \( \theta \)):

\( \cot(\theta) = \frac{1}{\tan(\theta)} \)

Since \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), we also have: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \)

In a right triangle: \( \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \)

Reciprocal Relationships - Summarized:

\( \csc(\theta) = \frac{1}{\sin(\theta)} \quad \Leftrightarrow \quad \sin(\theta) = \frac{1}{\csc(\theta)} \)

\( \sec(\theta) = \frac{1}{\cos(\theta)} \quad \Leftrightarrow \quad \cos(\theta) = \frac{1}{\sec(\theta)} \)

\( \cot(\theta) = \frac{1}{\tan(\theta)} \quad \Leftrightarrow \quad \tan(\theta) = \frac{1}{\cot(\theta)} \)

Mnemonic Tip: Notice the "co-" pairings: cosecant is reciprocal of sine, and cotangent is reciprocal of tangent. Secant (sec) is the reciprocal of cosine (cos) - the ones that *don't* have "co-" prefixes are reciprocals of each other.

3) Unit Circle and Reciprocal Functions 🌐

Let's revisit the unit circle to understand cosecant, secant, and cotangent in that context. Recall that for a point \( (x, y) = (\cos(\theta), \sin(\theta)) \) on the unit circle corresponding to angle \( \theta \):

\( \sin(\theta) = y \)
\( \cos(\theta) = x \)
\( \tan(\theta) = \frac{y}{x} \) (for \( x \neq 0 \))

Using these, we can express the reciprocal functions in terms of \( x \) and \( y \):

Cosecant: \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{y} \) (for \( y \neq 0 \))
Secant: \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{x} \) (for \( x \neq 0 \))
Cotangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{x}{y} \) (for \( y \neq 0 \))

Undefined Reciprocal Functions:

\( \csc(\theta) = \frac{1}{y} \) is undefined when \( y = 0 \), i.e., when \( \sin(\theta) = 0 \) (at \( 0^\circ, 180^\circ, 360^\circ, ... \) or \( 0, \pi, 2\pi, ... \) radians).
\( \sec(\theta) = \frac{1}{x} \) is undefined when \( x = 0 \), i.e., when \( \cos(\theta) = 0 \) (at \( 90^\circ, 270^\circ, ... \) or \( \frac{\pi}{2}, \frac{3\pi}{2}, ... \) radians).
\( \cot(\theta) = \frac{x}{y} \) is undefined when \( y = 0 \), i.e., when \( \sin(\theta) = 0 \) (same angles as where cosecant is undefined). Also note it's undefined when tangent is infinite, which also corresponds to \( \sin(\theta) = 0 \) and \( \cos(\theta) \neq 0 \). Actually, cotangent is undefined when \( \sin(\theta) = 0 \), and tangent is undefined when \( \cos(\theta) = 0 \).

4) Calculating Reciprocal Trigonometric Ratios - Examples 🎯

Example 1: Finding Reciprocal Ratios for a Special Angle

Find \( \csc(30^\circ), \sec(30^\circ), \cot(30^\circ) \).

Solution:

We know \( \sin(30^\circ) = \frac{1}{2}, \cos(30^\circ) = \frac{\sqrt{3}}{2}, \tan(30^\circ) = \frac{\sqrt{3}}{3} \).
\( \csc(30^\circ) = \frac{1}{\sin(30^\circ)} = \frac{1}{\frac{1}{2}} = 2 \)
\( \sec(30^\circ) = \frac{1}{\cos(30^\circ)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \) (rationalized)
\( \cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{\sqrt{3}}{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \) (rationalized)

Example 2: Finding Reciprocal Ratios Given Sine and Cosine

If \( \sin(\theta) = \frac{3}{5} \) and \( \cos(\theta) = \frac{4}{5} \), find \( \csc(\theta), \sec(\theta), \cot(\theta) \).

Solution:

\( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \)
\( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \)
\( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \) (Alternatively, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{3/5}{4/5} = \frac{3}{4} \), and \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{3/4} = \frac{4}{3} \)).

Example 3: All Six Ratios in a Right Triangle

Consider a right triangle with opposite side = 7, adjacent side = 24. First find the hypotenuse, then find all six trigonometric ratios for the angle \( \theta \) opposite the side of length 7.

Solution:

Hypotenuse \( = \sqrt{\text{Opposite}^2 + \text{Adjacent}^2} = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \).
\( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{25} \)
\( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{24}{25} \)
\( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{7}{24} \)
\( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{25}{7} \)
\( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{25}{24} \)
\( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{24}{7} \)

5) Practice Questions 🎯

5.1 Fundamental – Reciprocal Function Basics

1. Define cosecant, secant, and cotangent in terms of sine, cosine, and tangent.

2. What are the reciprocal ratios for sine, cosine, and tangent in terms of opposite, adjacent, and hypotenuse in a right triangle?

3. If \( \sin(\theta) = \frac{1}{3} \), what is \( \csc(\theta) \)?

4. If \( \cos(\theta) = \frac{\sqrt{2}}{2} \), what is \( \sec(\theta) \)?

5. If \( \tan(\theta) = \sqrt{3} \), what is \( \cot(\theta) \)?

6. What is \( \csc(45^\circ) \)? (Use the value of \( \sin(45^\circ) \)).

7. What is \( \sec(60^\circ) \)? (Use the value of \( \cos(60^\circ) \)).

8. What is \( \cot(30^\circ) \)? (Use the value of \( \tan(30^\circ) \)).

9. For what angles in the range \( 0^\circ \leq \theta < 360^\circ \) is \( \csc(\theta) \) undefined?

10. For what angles in the range \( 0^\circ \leq \theta < 360^\circ \) is \( \sec(\theta) \) undefined?

5.2 Challenging – Applications & Deeper Understanding 💪🚀

1. Given that \( \cot(\theta) = \frac{5}{12} \) and \( \sin(\theta) > 0 \) and \( \cos(\theta) < 0 \), find the values of all other five trigonometric ratios for angle \( \theta \). (Hint: Determine the quadrant of \( \theta \) first).

2. Simplify the expression: \( \sin(\theta) \csc(\theta) + \cos(\theta) \sec(\theta) \). Show your steps using reciprocal definitions.

3. Explain why \( \tan(\theta) \csc(\theta) = \sec(\theta) \cos(\theta) \) is NOT generally true, but \( \tan(\theta) \csc(\theta) = \sec(\theta) \sin(\theta) \) IS generally true. Correct the first equation to make it true.

4. Consider a right triangle where the ratio of the hypotenuse to the opposite side is 2. What is the angle opposite that side? (Use reciprocal ratio to find sine value).

5. (Conceptual) If you know the coordinates \( (x, y) \) of a point on the unit circle corresponding to an angle \( \theta \), explain how you would find \( \csc(\theta), \sec(\theta), \cot(\theta) \) directly using \( x \) and \( y \). What are the conditions for these to be defined?

6) Summary 🎉

Reciprocal Trig Functions: Cosecant (csc), Secant (sec), Cotangent (cot) are reciprocals of sine, cosine, tangent.
Definitions: \( \csc(\theta) = \frac{1}{\sin(\theta)}, \sec(\theta) = \frac{1}{\cos(\theta)}, \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} \).
Right Triangle Ratios: \( \csc(\theta) = \frac{\text{Hypotenuse}}{\text{Opposite}}, \sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}}, \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \).
Unit Circle: \( \csc(\theta) = \frac{1}{y}, \sec(\theta) = \frac{1}{x}, \cot(\theta) = \frac{x}{y} \), where \( (x, y) = (\cos(\theta), \sin(\theta)) \). Know when these are undefined.
Usage: Reciprocal functions complete the set of six trig ratios, simplify some formulas, historically used, and still relevant in various contexts.

Excellent! You've now expanded your trigonometric toolkit to include cosecant, secant, and cotangent! Understanding these reciprocal relationships is important for a complete foundation in trigonometry. Keep practicing with these ratios and you'll become comfortable using all six trigonometric functions! 🔄📐🌟