Secant, Cosecant and Cotangent - (2 of 4: Trigonometric Identities)

Maths Explained

17 Oct 201909:56

EducationalLearning

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TLDRThis video explores trigonometric identities involving cotangent, secant, and cosecant, which are not typically covered in GCSEs. It defines these functions as reciprocals of tangent, cosine, and sine, respectively. The video demonstrates how to derive and simplify identities using these functions, emphasizing the importance of breaking them down into sine and cosine for easier manipulation.

Takeaways

📚 The video discusses trigonometric identities involving cotangent, secant, and cosecant, which are not typically covered in GCSE but are important for A-levels.
🔍 Cotangent is defined as the reciprocal of tangent, or equivalently, cosine over sine.
🔍 Secant is defined as the reciprocal of cosine, and cosecant as the reciprocal of sine.
🔍 Key identities include: \(1 + \tan^2(\theta) = \sec^2(\theta)\) and \(1 + \cot^2(\theta) = \csc^2(\theta)\), both derived from the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\).
🔍 To derive these identities, divide the Pythagorean identity by \(\sin^2(\theta)\) and \(\cos^2(\theta)\) respectively.
🔍 An example identity is \(\cot(\theta) \csc(\theta) / (\sec^2(\theta) + \cos^2(\theta)) = \cos^3(\theta)\), which simplifies to \(\cos(\theta)\) when broken down into sine and cosine.
🔍 Another identity shown is \(\tan(\theta) + \cot(\theta) \times (\sin(\theta) + \cos(\theta)) = \sec(\theta) + \csc(\theta)\), simplified by converting tangent and cotangent into sine and cosine.
🔍 Simplifying complex expressions involving trigonometric functions often involves breaking them down into sine and cosine and then simplifying.
🔍 The video suggests that understanding these identities is crucial not just for proving them, but also for solving equations, differentiation, and integration.
🔍 The video is part of a series, with future content planned to cover more complex applications of these identities in calculus.

Q & A

What are the three trigonometric functions introduced in the video that are not commonly seen in GCSEs?
-The three trigonometric functions introduced in the video are cotangent (cot), secant (sec), and cosecant (csc).
How is cotangent defined in terms of sine and cosine?
-Cotangent is defined as the reciprocal of the tangent, which can be expressed as either 1 over tan(theta) or cosine over sine (cos(theta)/sin(theta)).
What is the definition of secant in terms of cosine?
-Secant is defined as the reciprocal of the cosine function, which is 1 over cos(theta).
How is cosecant related to the sine function?
-Cosecant is the reciprocal of the sine function, which is expressed as 1 over sin(theta).
What trigonometric identity involves 1 plus tan squared theta and sec squared theta?
-The identity is 1 plus tan squared theta equals sec squared theta (1 + tan^2(theta) = sec^2(theta)).
What is the identity that relates 1 plus cot squared theta to cosac squared theta?
-The identity is 1 plus cot squared theta equals cosac squared theta (1 + cot^2(theta) = csc^2(theta)).
Why is it useful to break trigonometric identities involving cot, sec, and csc down into sine and cosine?
-Breaking down identities into sine and cosine simplifies the expressions, making it easier to manipulate and simplify them, especially when proving identities or solving equations.
In the video, what identity is used to show that cot theta cosec theta divided by the sum of sec squared theta and cosine squared theta equals cos cubed theta?
-The identity used is cot(theta) * csc(theta) divided by (sec^2(theta) + cos^2(theta)) is equivalent to cos^3(theta).
What is the identity involving tan theta, cot theta, sine theta, and cosine theta that is shown in the video?
-The identity shown is tan(theta) + cot(theta) multiplied by (sine theta + cosine theta) is equivalent to sec(theta) + cos(theta).
Why is it recommended not to expand brackets immediately when proving trigonometric identities?
-Expanding brackets immediately can lead to a very messy expression, making it more difficult to simplify and prove the identity. It's often more effective to simplify the trigonometric functions first.
What are some of the other mathematical skills that one might need to apply when working with these trigonometric functions beyond proving identities?
-Beyond proving identities, one might need to solve equations, differentiate, and integrate functions involving these trigonometric functions.

Outlines

00:00

📚 Introduction to Trigonometric Identities

This paragraph introduces the topic of the video, which is to explore trigonometric identities, specifically focusing on less common functions such as cotangent, secant, and cosecant. The speaker clarifies that these functions are not typically covered in GCSEs but are not difficult to understand. Cotangent is defined as the reciprocal of the tangent, secant as the reciprocal of the cosine, and cosecant as the reciprocal of the sine. The paragraph also introduces two identities involving these functions, which are derived from the Pythagorean identity, and sets the stage for further discussion on additional identities.

05:04

🔍 Demonstrating Trigonometric Identities

The second paragraph delves into the demonstration of trigonometric identities involving the newly introduced functions. The speaker provides a step-by-step guide on how to simplify and prove the identity involving cotangent, secant, and cosine. The process involves breaking down the expression into sine and cosine components, simplifying the fractions, and eventually arriving at the identity's simplified form. The paragraph also touches on the importance of being able to handle these functions in various mathematical contexts, such as solving equations and performing calculus operations, hinting at the possibility of covering these topics in future videos.

Mindmap

Keywords

💡Trigonometric Identities

Trigonometric identities are mathematical equations that hold true for all values of the variable involved, typically within the domain of trigonometry. In the video, identities are used to simplify and understand the relationships between trigonometric functions like sine, cosine, and their reciprocals. For example, the identity 1 + tan^2(θ) = sec^2(θ) is mentioned, which is derived from the Pythagorean identity and is central to the theme of the video.

💡Cotangent

Cotangent, denoted as cot(θ), is the reciprocal of the tangent function, which is defined as the ratio of adjacent to opposite in a right-angled triangle, or cosine over sine. In the script, cotangent is introduced as a function that is not commonly seen in GCSEs but is essential for understanding more advanced trigonometric concepts, such as the identity cot(θ) = cos(θ)/sin(θ).

💡Secant

Secant, represented as sec(θ), is the reciprocal of the cosine function. It is another advanced trigonometric function that extends the basic sine and cosine functions. The script explains secant as 1 over the cosine, or the reciprocal of the cosine function, which is essential for deriving and understanding certain trigonometric identities.

💡Cosecant

Cosecant, noted as csc(θ), is the reciprocal of the sine function. It is introduced in the video as a function that is not typically covered in GCSEs but is important for A-levels and beyond. The script uses the definition of cosecant as 1 over sine, which is crucial for understanding the identity 1 + cot^2(θ) = csc^2(θ).

💡Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity stating that sine squared plus cosine squared equals 1, i.e., sin^2(θ) + cos^2(θ) = 1. This identity is the basis for deriving other identities in the video, such as those involving secant and cosecant, and is central to the theme of the video.

💡Reciprocal Functions

Reciprocal functions in trigonometry are those functions that are the inverse of the basic sine and cosine functions, such as cotangent, secant, and cosecant. The video script explains these reciprocal functions as 1 over their respective trigonometric functions, which is essential for understanding more complex trigonometric identities and equations.

💡Sine and Cosine

Sine and cosine are the two primary trigonometric functions that relate the angles of a right-angled triangle to the lengths of its sides. In the video, sine and cosine are the basis for defining other trigonometric functions and identities, such as cotangent being the reciprocal of tangent, which is sine over cosine.

💡Tangent

Tangent, abbreviated as tan(θ), is a trigonometric function that represents the ratio of the opposite side to the adjacent side in a right-angled triangle, or sine over cosine. The script uses tangent in the context of defining cotangent and in the derivation of identities, such as tan(θ) + cot(θ) = sec(θ) + csc(θ).

💡Simplification

Simplification in the context of the video refers to the process of breaking down complex trigonometric expressions into simpler forms, typically involving sine and cosine, to make them easier to understand and manipulate. The script demonstrates simplification in the process of proving identities, such as expressing cotangent and cosecant in terms of sine and cosine before simplifying.

💡Proof

A proof in mathematics is a demonstration that a statement is true. In the video, the script involves proving trigonometric identities by showing that both sides of an equation are equal through logical steps and manipulation of trigonometric functions. Examples from the script include proving that cot(θ)cosec(θ) / (sec^2(θ) + cos^2(θ)) is equivalent to cos^3(θ).

💡Differentiation and Integration

Differentiation and integration are fundamental operations in calculus, which are briefly mentioned in the script as part of the broader application of understanding trigonometric functions. While not the main focus of the video, these concepts are implied as further steps in the study of trigonometry beyond identities.

Highlights

Introduction of trigonometric identities involving cotangent, secant, and cosecant functions.

Cotangent defined as the reciprocal of tangent, or cosine over sine.

Secant as the reciprocal of cosine, and cosecant as the reciprocal of sine.

Explanation of identities derived from the Pythagorean identity: sine squared plus cosine squared equals 1.

Demonstration of the identity 1 + tan squared theta equals sec squared theta.

Presentation of the identity 1 + cot squared theta equals cosec squared theta.

Methodology to simplify complex trigonometric expressions by breaking them down into sine and cosine.

Example identity proof: cot theta cosec theta divided by sec squared theta plus cosine squared theta equals cos cubed theta.

Technique of simplifying the numerator and denominator separately for complex expressions.

Cancellation of terms in trigonometric expressions to reach simplified results.

Second identity proof involving tan theta plus cot theta multiplied by sine theta plus cosine theta equals sec theta plus cosine theta.

Strategy to avoid expanding messy brackets by simplifying expressions first.

Combining fractions and cross-multiplying to simplify trigonometric expressions.

Transformation of trigonometric expressions to reveal the secant and cosecant functions.

Final simplification leading to the proof of the identity involving secant and cosecant.

Emphasis on the importance of understanding and applying these identities in solving equations, differentiation, and integration.

Teaser for future content on solving equations and calculus problems involving these trigonometric identities.

Encouragement for viewers to like, subscribe, and look forward to more content on this topic.

Transcripts

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Trig Integral Practice | MIT 18.01SC Single Variable Calculus, Fall 2010

Lesson 1 - Basic Trig Identities Involving Sin, Cos, and Tan

Secant, Cosecant and Cotangent - (2 of 4: Trigonometric Identities)

Takeaways

Q & A

What are the three trigonometric functions introduced in the video that are not commonly seen in GCSEs?

How is cotangent defined in terms of sine and cosine?

What is the definition of secant in terms of cosine?

How is cosecant related to the sine function?

What trigonometric identity involves 1 plus tan squared theta and sec squared theta?

What is the identity that relates 1 plus cot squared theta to cosac squared theta?

Why is it useful to break trigonometric identities involving cot, sec, and csc down into sine and cosine?

In the video, what identity is used to show that cot theta cosec theta divided by the sum of sec squared theta and cosine squared theta equals cos cubed theta?

What is the identity involving tan theta, cot theta, sine theta, and cosine theta that is shown in the video?

Why is it recommended not to expand brackets immediately when proving trigonometric identities?

What are some of the other mathematical skills that one might need to apply when working with these trigonometric functions beyond proving identities?