Trigonometric Ratios Of Complementary Angles

In trigonometry, complementary angles are two angles whose measures add up to 90 degrees, and their sine and cosine values are interrelated. Specifically, the sine of one angle equals the cosine of its complement: sin(θ) = cos(90°-θ), and vice versa. Remembering this relationship can simplify solving many trigonometric problems.

Get started Sign up for free
Trigonometric Ratios Of Complementary Angles Trigonometric Ratios Of Complementary Angles

Create learning materials about Trigonometric Ratios Of Complementary Angles with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account

Millions of flashcards designed to help you ace your studies

Sign up for free

Convert documents into flashcards for free with AI!

Contents
Table of contents

    Trigonometric Ratios Of Complementary Angles

    When exploring the world of trigonometry, understanding the relationships and properties of angles is crucial. One particularly interesting property is how trigonometric ratios change when dealing with complementary angles.

    Definition

    Complementary angles are two angles whose sum is 90 degrees. In mathematical terms, if you have angles A and B, they are complementary if:

    \[ A + B = 90^\circ \]

    For instance, if angle A is 30 degrees, the angle complementary to A is 60 degrees, because \(30^\circ + 60^\circ = 90^\circ\).

    When it comes to trigonometric functions such as sine, cosine, and tangent, they exhibit unique relationships when working with complementary angles. This understanding can simplify solving various trigonometric problems.

    Trigonometric Identities for Complementary Angles

    There are three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). For complementary angles say, \( \theta \) and \( 90^\circ - \theta \), there are specific identities:

    Sine and Cosine:

    The sine of an angle is equal to the cosine of its complementary angle. Mathematically, this is represented as:

    \[ \sin(\theta) = \cos( 90^\circ - \theta ) \]

    Similarly,

    \[ \cos(\theta) = \sin( 90^\circ - \theta ) \]

    Tangent and Cotangent:

    The tangent of an angle is equal to the cotangent of its complementary angle. This can be expressed as:

    \[ \tan(\theta) = \cot( 90^\circ - \theta ) \]

    where \( \cot \theta \) is the cotangent function.

    For instance, consider an angle \(\theta = 40^\circ\). The complementary angle is \(90^\circ - 40^\circ = 50^\circ\).

    According to the identities:

    • \( \sin(40^\circ) = \cos(50^\circ) \)
    • \( \cos(40^\circ) = \sin(50^\circ) \)
    • \( \tan(40^\circ) = \cot(50^\circ) \)

    Knowing these identities simplifies computations and helps in verifying results without having to re-calculate trigonometric functions every time.

    Applications of These Identities

    These trigonometric identities have practical applications in various fields such as physics, engineering, and architecture. They help in:

    • Solving right triangle problems more easily.
    • Reducing the complexity of trigonometric equations by substituting one ratio with another.
    • Understanding the symmetry and behaviour of trigonometric functions.

    A deeper dive into these identities reveals interesting properties about trigonometric functions:

    Reciprocal Identities: Each of the primary trigonometric functions has a reciprocal identity:

    • \( \sin \theta = \frac{1}{\csc \theta} \)
    • \( \cos \theta = \frac{1}{\sec \theta} \)
    • \( \tan \theta = \frac{1}{\cot \theta} \)

    These identities are important when solving complex trigonometric equations and for expressing the functions in alternative forms for analytical convenience.

    Remember that understanding these relationships makes analysing trigonometric functions more intuitive and less calculation-intensive.

    Trigonometric Ratios Of Complementary Angles

    When exploring the world of trigonometry, understanding the relationships and properties of angles is crucial. One particularly interesting property is how trigonometric ratios change when dealing with complementary angles.

    Definition

    Complementary angles are two angles whose sum is 90 degrees. In mathematical terms, if you have angles A and B, they are complementary if:

    \[ A + B = 90^\circ \]

    For instance, if angle A is 30 degrees, the angle complementary to A is 60 degrees, because \(30^\circ + 60^\circ = 90^\circ\).

    When it comes to trigonometric functions such as sine, cosine, and tangent, they exhibit unique relationships when working with complementary angles. This understanding can simplify solving various trigonometric problems.

    Trigonometric Identities for Complementary Angles

    There are three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). For complementary angles say, \( \theta \) and \( 90^\circ - \theta \), there are specific identities:

    Sine and Cosine:

    The sine of an angle is equal to the cosine of its complementary angle. Mathematically, this is represented as:

    \[ \sin(\theta) = \cos( 90^\circ - \theta ) \]

    Similarly,

    \[ \cos(\theta) = \sin( 90^\circ - \theta ) \]

    Tangent and Cotangent:

    The tangent of an angle is equal to the cotangent of its complementary angle. This can be expressed as:

    \[ \tan(\theta) = \cot( 90^\circ - \theta ) \]

    where \( \cot \theta \) is the cotangent function.

    For instance, consider an angle \(\theta = 40^\circ\). The complementary angle is \(90^\circ - 40^\circ = 50^\circ\).

    According to the identities:

    • \( \sin(40^\circ) = \cos(50^\circ) \)
    • \( \cos(40^\circ) = \sin(50^\circ) \)
    • \( \tan(40^\circ) = \cot(50^\circ) \)

    Knowing these identities simplifies computations and helps in verifying results without having to re-calculate trigonometric functions every time.

    Applications of These Identities

    These trigonometric identities have practical applications in various fields such as physics, engineering, and architecture. They help in:

    • Solving right triangle problems more easily.
    • Reducing the complexity of trigonometric equations by substituting one ratio with another.
    • Understanding the symmetry and behaviour of trigonometric functions.

    A deeper dive into these identities reveals interesting properties about trigonometric functions:

    Reciprocal Identities: Each of the primary trigonometric functions has a reciprocal identity:

    • \( \sin \theta = \frac{1}{\csc \theta} \)
    • \( \cos \theta = \frac{1}{\sec \theta} \)
    • \( \tan \theta = \frac{1}{\cot \theta} \)

    These identities are important when solving complex trigonometric equations and for expressing the functions in alternative forms for analytical convenience.

    Remember that understanding these relationships makes analysing trigonometric functions more intuitive and less calculation-intensive.

    Trigonometric Ratios Of Complementary Angles

    When exploring the world of trigonometry, understanding the relationships and properties of angles is crucial. One particularly interesting property is how trigonometric ratios change when dealing with complementary angles.

    Definition

    Complementary angles are two angles whose sum is 90 degrees. In mathematical terms, if you have angles A and B, they are complementary if:

    \[ A + B = 90^\circ \]

    For instance, if angle A is 30 degrees, the angle complementary to A is 60 degrees, because \(30^\circ + 60^\circ = 90^\circ\).

    When it comes to trigonometric functions such as sine, cosine, and tangent, they exhibit unique relationships when working with complementary angles. This understanding can simplify solving various trigonometric problems.

    Trigonometric Identities for Complementary Angles

    There are three primary trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). For complementary angles say, \( \theta \) and \( 90^\circ - \theta \), there are specific identities:

    Sine and Cosine:

    The sine of an angle is equal to the cosine of its complementary angle. Mathematically, this is represented as:

    \[ \sin(\theta) = \cos( 90^\circ - \theta ) \]

    Similarly,

    \[ \cos(\theta) = \sin( 90^\circ - \theta ) \]

    Tangent and Cotangent:

    The tangent of an angle is equal to the cotangent of its complementary angle. This can be expressed as:

    \[ \tan(\theta) = \cot( 90^\circ - \theta ) \]

    where \( \cot \theta \) is the cotangent function.

    For instance, consider an angle \(\theta = 40^\circ\). The complementary angle is \(90^\circ - 40^\circ = 50^\circ\).

    According to the identities:

    • \( \sin(40^\circ) = \cos(50^\circ) \)
    • \( \cos(40^\circ) = \sin(50^\circ) \)
    • \( \tan(40^\circ) = \cot(50^\circ) \)

    Knowing these identities simplifies computations and helps in verifying results without having to re-calculate trigonometric functions every time.

    Applications of These Identities

    These trigonometric identities have practical applications in various fields such as physics, engineering, and architecture. They help in:

    • Solving right triangle problems more easily.
    • Reducing the complexity of trigonometric equations by substituting one ratio with another.
    • Understanding the symmetry and behaviour of trigonometric functions.

    A deeper dive into these identities reveals interesting properties about trigonometric functions:

    Reciprocal Identities: Each of the primary trigonometric functions has a reciprocal identity:

    • \( \sin \theta = \frac{1}{\csc \theta} \)
    • \( \cos \theta = \frac{1}{\sec \theta} \)
    • \( \tan \theta = \frac{1}{\cot \theta} \)

    These identities are important when solving complex trigonometric equations and for expressing the functions in alternative forms for analytical convenience.

    Remember that understanding these relationships makes analysing trigonometric functions more intuitive and less calculation-intensive.

    Proof Of Trigonometric Ratios Of Complementary Angles

    In trigonometry, the relationship between trigonometric ratios and complementary angles is essential for simplifying and solving various problems. This section will provide a detailed proof of these relationships, which can help solidify your understanding of the concepts.

    Understanding Trigonometric Ratios Of Complementary Angles

    Complementary angles are two angles whose sum is 90 degrees. If you have angles A and B, they are complementary if:

    \[ A + B = 90^\circ \]

    For example, if angle A is 30 degrees, the angle complementary to A is 60 degrees, because \(30^\circ + 60^\circ = 90^\circ\).

    To understand the trigonometric ratios of complementary angles better, consider the primary trigonometric identities:

    • The sine of an angle is equal to the cosine of its complementary angle: \(\sin(\theta) = \cos(90^\circ - \theta)\).
    • The cosine of an angle is equal to the sine of its complementary angle: \(\cos(\theta) = \sin(90^\circ - \theta)\).
    • The tangent of an angle is equal to the cotangent of its complementary angle: \(\tan(\theta) = \cot(90^\circ - \theta)\).

    Consider an angle \(\theta = 45^\circ\). The complementary angle is \(90^\circ - 45^\circ = 45^\circ\). According to the identities:

    • \( \sin(45^\circ) = \cos(45^\circ) \)
    • \( \cos(45^\circ) = \sin(45^\circ) \)
    • \( \tan(45^\circ) = \cot(45^\circ) \)

    A deeper dive into these identities reveals interesting properties about trigonometric functions:

    Reciprocal Identities: Each of the primary trigonometric functions has a reciprocal identity:

    • \( \sin \theta = \frac{1}{\csc \theta} \)
    • \( \cos \theta = \frac{1}{\sec \theta} \)
    • \( \tan \theta = \frac{1}{\cot \theta} \)

    These identities are important when solving complex trigonometric equations and for expressing the functions in alternative forms for analytical convenience.

    How To Solve Trigonometric Ratios Of Complementary Angles

    Solving trigonometric ratios of complementary angles involves applying the identities mentioned earlier. Let's go over the steps:

    Suppose you need to find the cosine of a 30-degree angle. Using the identity for complementary angles, you can say:

    \[ \cos(30^\circ) = \sin(90^\circ - 30^\circ)\]

    Since \(90^\circ - 30^\circ = 60^\circ\), it becomes:

    \[ \cos(30^\circ) = \sin(60^\circ)\]

    This simplifies the calculation process significantly.

    To solve problems efficiently, follow these steps:

    • Identify the given angle and its complementary angle.
    • Apply the appropriate trigonometric identity.
    • Calculate or use known trigonometric values to find the desired ratio.

    Remember that understanding these relationships makes analysing trigonometric functions more intuitive and less calculation-intensive.

    Trigonometric Ratios Of Complementary Angles - Key takeaways

    • Definition: Complementary angles are two angles whose sum is 90 degrees. Mathematically, if angles A and B are such that A + B = 90°, they are complementary.
    • Trigonometric Identities for Complementary Angles: The sine of an angle is equal to the cosine of its complementary angle and vice versa. For example, sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). The tangent of an angle is equal to the cotangent of its complementary angle: tan(θ) = cot(90° - θ).
    • Proof of Trigonometric Ratios of Complementary Angles: These relationships can be proven and simplify solving trigonometric problems, for instance, sin(40°) = cos(50°).
    • Reciprocal Identities: Each primary trigonometric function has a reciprocal identity: sin(θ) = 1/csc(θ), cos(θ) = 1/sec(θ), tan(θ) = 1/cot(θ).
    • Applications and Solving Strategies: These identities are used in solving right triangle problems, simplifying trigonometric equations, and understanding the symmetric behaviour of trigonometric functions. Steps include identifying the given angle, using the trigonometric identity, and calculating the desired ratio.
    Frequently Asked Questions about Trigonometric Ratios Of Complementary Angles
    What are the trigonometric ratios of complementary angles?
    The trigonometric ratios of complementary angles state that sin(90° - θ) = cos(θ), cos(90° - θ) = sin(θ), tan(90° - θ) = cot(θ), cot(90° - θ) = tan(θ), sec(90° - θ) = cosec(θ), and cosec(90° - θ) = sec(θ).
    How are the trigonometric ratios of complementary angles derived?
    The trigonometric ratios of complementary angles are derived from the fact that the sum of two complementary angles is 90 degrees. Therefore, for any angle θ, sin(90° - θ) = cos(θ) and cos(90° - θ) = sin(θ), likewise for the other ratios: tan(90° - θ) = cot(θ) and cot(90° - θ) = tan(θ).
    Why are the trigonometric ratios of complementary angles significant in geometry?
    The trigonometric ratios of complementary angles are significant in geometry because they reveal the relationships between sine, cosine, and other functions for angles summing to 90°, simplifying calculations and proofs. This property supports problem-solving in right-angled triangles and applications in various geometric contexts.
    What is the relationship between the trigonometric ratios of complementary angles?
    The trigonometric ratios of complementary angles are related as follows: the sine of an angle equals the cosine of its complement, the cosine of an angle equals the sine of its complement, the tangent of an angle equals the cotangent of its complement, and the secant of an angle equals the cosecant of its complement.
    How do the trigonometric ratios of complementary angles apply in real-life scenarios?
    Trigonometric ratios of complementary angles are used in real-life scenarios such as engineering, architecture, and physics to calculate heights, distances, and angles accurately. This is crucial for construction, navigation, and designing various mechanical systems. Additionally, they aid in solving problems related to wave behaviour and circuit analysis.

    Test your knowledge with multiple choice flashcards

    Which trigonometric identity is true for the tangent of an angle and its complementary angle?

    What is the definition of complementary angles?

    What are complementary angles?

    Next

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 10 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App
    Sign up with Email

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner