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TRIGONOMETRIC RATIOS OF ALLIED AND COMPOUND ANGLES|FIRST YEARDIPLOMA|POLYTECHNIC|Lecture 01|K Scheme

Pradeep Giri Academy・40 minutes read

The session covers fundamental trigonometry concepts for diploma and polytechnic students, focusing on compound angle formulas and the relationships between trigonometric functions in different quadrants. It emphasizes the importance of understanding the signs of these functions based on angle positions and transformations to solve trigonometric problems effectively.

Insights

  • The session aims to demystify trigonometry for diploma and polytechnic students by addressing common anxieties about formulas and concepts, particularly focusing on compound angles and their relationships, which are essential for accurate problem-solving in various quadrants.
  • Key formulas for compound angles, such as Sin(A + B) and Cos(A + B), are introduced along with their applications and transformations across different quadrants, highlighting the importance of understanding the signs of trigonometric functions and the relationships between angles to effectively tackle trigonometric calculations.

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Recent questions

  • What is trigonometry used for?

    Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right triangles. It is widely used in various fields such as physics, engineering, astronomy, and architecture. Trigonometric functions like sine, cosine, and tangent help in solving problems involving angles and distances. For instance, in physics, trigonometry is essential for analyzing wave patterns and oscillations, while in engineering, it aids in designing structures and understanding forces. Additionally, trigonometry plays a crucial role in navigation and computer graphics, making it a fundamental tool in both theoretical and applied sciences.

  • How do you calculate sine and cosine?

    To calculate sine and cosine, you typically use the definitions based on a right triangle or the unit circle. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse. For example, if you have a right triangle with an angle θ, the sine is calculated as sin(θ) = opposite/hypotenuse, and cosine as cos(θ) = adjacent/hypotenuse. Alternatively, on the unit circle, the sine of an angle corresponds to the y-coordinate, and the cosine corresponds to the x-coordinate of a point on the circle. These relationships are fundamental in trigonometry and are used to derive various trigonometric identities and formulas.

  • What are the main trigonometric functions?

    The main trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are defined based on the ratios of the sides of a right triangle. Sine is the ratio of the length of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. Additionally, there are three reciprocal functions: cosecant (cosec), secant (sec), and cotangent (cot), which are defined as the reciprocals of sine, cosine, and tangent, respectively. These functions are essential for solving various mathematical problems involving angles and are widely used in calculus, physics, and engineering.

  • What is a compound angle in trigonometry?

    A compound angle in trigonometry refers to an angle formed by the sum or difference of two angles. For example, if A and B are two angles, then A + B and A - B are considered compound angles. The significance of compound angles lies in their ability to simplify the calculation of trigonometric functions for these angles using specific formulas. For instance, the sine of a compound angle can be expressed as sin(A + B) = sinA * cosB + cosA * sinB, while the cosine can be expressed as cos(A + B) = cosA * cosB - sinA * sinB. These formulas are crucial for solving complex trigonometric problems and are widely used in various applications, including physics and engineering.

  • How do you remember trigonometric identities?

    Remembering trigonometric identities can be made easier through the use of mnemonic devices and practice. One popular mnemonic for remembering which trigonometric functions are positive in each quadrant is "All Students Take Calculus," where each word corresponds to the functions that are positive in the first, second, third, and fourth quadrants, respectively. Additionally, regularly practicing the derivation and application of these identities helps reinforce memory. Creating flashcards with different identities and their corresponding formulas can also be beneficial. Engaging in problem-solving exercises that require the use of these identities will further enhance understanding and retention, making it easier to recall them when needed.

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Summary

00:00

Understanding Trigonometry for Students

  • The session focuses on trigonometry basics, targeting diploma and polytechnic students, addressing common fears related to formulas and concepts in the subject.
  • The discussion begins with the fifth chapter, covering Trigonometry Ratios of Allied and Compound Angles, emphasizing acute and compound angles.
  • Compound angles are defined as the algebraic sum or difference of two angles, requiring at least two angles for calculations.
  • Six essential formulas for compound angles are introduced, highlighting the importance of remembering the relationships between sine, cosine, and tangent functions.
  • The formula for the sum of two angles, A + B, is given as Sin(A + B) = SinA * CosB + CosA * SinB.
  • For subtraction, the formula is Sin(A - B) = SinA * CosB - CosA * SinB, emphasizing the change in sign for subtraction.
  • Cosine formulas are presented, with Cos(A + B) = CosA * CosB - SinA * SinB and Cos(A - B) = CosA * CosB + SinA * SinB.
  • The tangent formulas are outlined as Tan(A + B) = (TanA + TanB) / (1 - TanA * TanB) and Tan(A - B) = (TanA - TanB) / (1 + TanA * TanB).
  • The session also covers light angles, defined as angles that are integral multiples of 90 degrees, with specific values like 0, π/2, and 180 degrees discussed.
  • The coordinate system is explained, detailing the quadrants and angle measurements, reinforcing the relationship between degrees and radians, such as 90° = π/2.

13:35

Understanding Angles and Trigonometric Functions

  • A semicircle is completed by moving downwards, reaching 270°, which corresponds to 3π/2 radians, indicating a downward position in the circle.
  • The full circle measures 360°, equivalent to 2π radians, starting from 0 and returning to 2π after one complete rotation.
  • Completing two full circles results in 4π radians, while a single complete circle is represented as 2π, confirming the relationship between degrees and radians.
  • The movement direction affects angle signs: moving anti-clockwise is positive, while clockwise movement is negative, impacting trigonometric calculations.
  • The first quadrant contains all positive trigonometric functions, while the second quadrant has only sine and cosecant as positive, with the rest negative.
  • In the third quadrant, tangent and cotangent are positive, while sine, cosine, secant, and cosecant are negative.
  • The fourth quadrant has positive cosine and secant, with sine, tangent, cosecant, and cotangent being negative.
  • To remember trigonometric signs, use mnemonic devices like "All Students Take Calculus," indicating which functions are positive in each quadrant.
  • When dealing with angles like π/2 ± θ, the corresponding trigonometric values must be adjusted according to quadrant rules and sign conventions.
  • Understanding the relationships between angles and their trigonometric values is crucial for solving problems accurately, especially in different quadrants.

27:13

Trigonometric Functions and Quadrant Sign Rules

  • The expression 3π/2 + θ indicates a position in the fourth quadrant, where cosecant (cosec) and secant (sec) are positive, while sine, cosine, and tangent are negative.
  • For angles like π/2 or 4π/2, if n is a multiple of 5, adjustments must be made using trigonometric identities to maintain correct values.
  • In the fourth quadrant, the transformations are: sinθ becomes -sinθ, tanθ changes to -cotθ, cosecθ becomes secθ, and secθ changes to cosecθ.
  • The expression 3π/2 - θ indicates a position in the third quadrant, where tangent (tan) and cotangent (cot) are positive, while sine, cosecant, and secant are negative.
  • For angles like π + θ, cotangent remains positive, while sine, cosecant, and secant are negative, indicating no transformations are needed for these ratios.
  • When dealing with angles like 2π + θ, all trigonometric functions remain positive, as this angle is in the first quadrant.
  • The basic trigonometric values to remember are: sin(0) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, and sin(90°) = 1.
  • The reciprocal relationships for trigonometric functions are: cosecθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ.
  • The compound angle formula for sin(π/2 + θ) is derived as sin(π/2)cos(θ) + cos(π/2)sin(θ), simplifying to sin(θ) since sin(π/2) = 1 and cos(π/2) = 0.
  • Understanding the quadrant system and the signs of trigonometric functions is crucial for solving problems involving multiple angles and transformations in trigonometry.

43:05

Understanding Compound Angles and Formulas

  • The relationship between compound angles is explored using the formula cos(A + B) = cosA cosB - sinA sinB, emphasizing practice with formulas for better understanding and retention.
  • Key values include sin(0) = 0, sin(π) = 0, cos(0) = 1, and cos(π) = -1, with sessions planned for further practice and application of these concepts.
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