Glossary

7.2 Sum and Difference Identities

2 min read•july 25, 2024

Trigonometric are powerful tools for simplifying complex expressions and solving equations. These formulas allow us to break down compound angles into simpler components, making calculations easier and more manageable.

By mastering these identities, we gain a deeper understanding of how trigonometric functions behave when angles are added or subtracted. This knowledge is crucial for tackling advanced problems in trigonometry and preparing for more advanced mathematical concepts.

Deriving and Applying Sum and Difference Identities

Sum and difference formulas

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  • Geometric approach employs unit circle and angle addition properties visualizing angle addition on coordinate plane
  • sum formula cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B leads to cosine difference formula
  • sum formula sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B generates sine difference formula
  • sum formula tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} derived using quotient identity and previous formulas
  • Tangent difference formula follows similar derivation process

Application of trigonometric identities

  • Recognize expressions sin⁡(A±B)\sin(A ± B), cos⁡(A±B)\cos(A ± B), or tan⁡(A±B)\tan(A ± B) for identity application
  • Cosine formulas: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A \cos B - \sin A \sin B, cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A \cos B + \sin A \sin B
  • Sine formulas: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B, sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B
  • Tangent formulas: tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}, tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
  • Evaluate expressions with special angles (30°, 45°, 60°) or use calculator for non-standard values

Simplification with identities

  • Break down compound angles into sums or differences
  • Combine like terms after applying identities
  • Factor common terms and rationalize denominators when necessary
  • Verify simplifications using other trigonometric identities
  • Apply sum and difference identities alongside other trig identities for comprehensive simplification

Solving equations with identities

  • Rewrite equations using appropriate sum or difference identity
  • Isolate variable through algebraic techniques (combining terms, factoring)
  • Solve using inverse trigonometric functions
  • Determine general solution and restrict to specific interval if needed
  • Verify solutions by substitution into original equation
  • Tackle equations with multiple angles by rewriting in terms of a single angle using sum and difference identities

Key Terms to Review (15)

√3: The term √3 refers to the square root of three, which is an important irrational number approximately equal to 1.732. It plays a significant role in various mathematical contexts, especially in trigonometry, where it appears in the values of sine and cosine for certain angles, as well as in relationships derived from the sum and difference identities. Understanding √3 helps in simplifying expressions and solving trigonometric equations.
30 degrees: 30 degrees is a specific angle measurement commonly encountered in trigonometry, representing one-sixth of a full rotation. This angle is fundamental in various trigonometric calculations, allowing for the evaluation of sine, cosine, and tangent values, as well as serving as a building block for understanding angles in different contexts such as the unit circle and right triangles.
45 degrees: 45 degrees is an angle that is exactly halfway between 0 degrees and 90 degrees, making it one of the most commonly encountered angles in trigonometry. This angle is significant because it has specific sine, cosine, and tangent values, and it plays a crucial role in understanding the behavior of trigonometric functions in various contexts.
60 degrees: 60 degrees is a specific angle measure that is significant in trigonometry, especially in the study of triangles and circular functions. It is one of the angles in an equilateral triangle, where all angles are equal, and each measures 60 degrees. Understanding 60 degrees is crucial for calculating sine, cosine, and tangent values, as well as for applying sum and difference identities in trigonometric equations.
Complementary Angles: Complementary angles are two angles whose measures add up to exactly 90 degrees. This concept is crucial when dealing with various trigonometric functions and identities, as complementary angles have specific relationships that can simplify calculations and deepen understanding of angle measures in different contexts.
Cosine: Cosine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. This relationship holds true in various contexts, from defining circular functions on the unit circle to solving complex triangles, making it essential in understanding angles and their measures.
Pythagorean Identity: The Pythagorean Identity is a fundamental relationship in trigonometry that expresses the square of the sine function plus the square of the cosine function as equal to one, represented as $$ ext{sin}^2(x) + ext{cos}^2(x) = 1$$. This identity is crucial in connecting the concepts of angle measures and the unit circle, forming a basis for deriving other identities and solving various trigonometric problems.
Quadrant: A quadrant refers to one of the four sections of the Cartesian coordinate plane, divided by the x-axis and y-axis. Each quadrant has specific characteristics related to the signs of the coordinates and plays a vital role in understanding the behavior of trigonometric functions. The relationship between quadrants and trigonometric values is crucial for analyzing transformations and applying sum and difference identities.
Reference Angle: A reference angle is the smallest angle formed by the terminal side of a given angle and the x-axis, always measured as a positive acute angle. This concept is crucial for understanding how angles relate to trigonometric functions across different quadrants, as it helps simplify calculations by connecting them to familiar angles in the first quadrant.
Sin(a ± b): The term sin(a ± b) refers to the sine of the sum or difference of two angles, a and b. This identity is part of a set of formulas that allow for the simplification and calculation of sine values when dealing with combined angles. These identities are crucial in trigonometry as they enable the breakdown of complex angle measures into more manageable components, aiding in various applications like solving triangles and analyzing periodic functions.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. This relationship extends beyond right triangles to various contexts, such as the unit circle and periodic functions, where sine represents the y-coordinate of a point on the unit circle corresponding to a given angle.
Sum and Difference Identities: Sum and Difference Identities are mathematical formulas that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sines and cosines of the individual angles. These identities help simplify complex trigonometric expressions and enable the calculation of angles that are not found on the unit circle. They are essential tools in trigonometry for solving equations, proving other identities, and analyzing functions.
Tan(a ± b): The term tan(a ± b) refers to the tangent of the sum or difference of two angles, a and b. This identity is part of a larger set of trigonometric identities that allow for the expression of the tangent of combined angles in terms of the tangents of the individual angles. These identities are crucial for simplifying expressions and solving equations that involve trigonometric functions.
Tangent: The tangent function, often abbreviated as 'tan', relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. This function is crucial in various branches of mathematics, especially in solving triangles and modeling periodic phenomena.
π: π (pi) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter, approximately equal to 3.14159. This irrational number is crucial in various mathematical contexts, especially in trigonometry, where it serves as a foundation for understanding angles and periodic functions.
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