Sum and difference identities are powerful tools for manipulating trigonometric expressions. They allow us to break down complex angle combinations into simpler terms, making calculations easier and revealing hidden relationships between angles.
These identities are crucial for solving advanced trigonometric equations and simplifying complex expressions. They're the building blocks for understanding more complex trigonometric concepts and are widely used in fields like physics and engineering.
Sum and Difference Identities
Sum and difference formulas
Top images from around the web for Sum and difference formulas
Sum and Difference Identities – Algebra and Trigonometry OpenStax View original
Is this image relevant?
TrigCheatSheet.com: Graphing Sine, Cosine, and Tangent View original
Is this image relevant?
TrigCheatSheet.com: Graphing Sine, Cosine, and Tangent View original
Is this image relevant?
Sum and Difference Identities – Algebra and Trigonometry OpenStax View original
Is this image relevant?
TrigCheatSheet.com: Graphing Sine, Cosine, and Tangent View original
Is this image relevant?
1 of 3
Top images from around the web for Sum and difference formulas
Sum and Difference Identities – Algebra and Trigonometry OpenStax View original
Is this image relevant?
TrigCheatSheet.com: Graphing Sine, Cosine, and Tangent View original
Is this image relevant?
TrigCheatSheet.com: Graphing Sine, Cosine, and Tangent View original
Is this image relevant?
Sum and Difference Identities – Algebra and Trigonometry OpenStax View original
Is this image relevant?
TrigCheatSheet.com: Graphing Sine, Cosine, and Tangent View original
Is this image relevant?
1 of 3
Sum formula for sine expresses sin(A+B) in terms of sinA, cosA, sinB, and cosB: sin(A+B)=sinAcosB+cosAsinB
Difference formula for sine expresses sin(A−B) in terms of sinA, cosA, sinB, and cosB: sin(A−B)=sinAcosB−cosAsinB
Sum formula for cosine expresses cos(A+B) in terms of cosA, cosB, sinA, and sinB: cos(A+B)=cosAcosB−sinAsinB
Difference formula for cosine expresses cos(A−B) in terms of cosA, cosB, sinA, and sinB: cos(A−B)=cosAcosB+sinAsinB
Sum formula for tangent expresses tan(A+B) in terms of tanA and tanB: tan(A+B)=1−tanAtanBtanA+tanB
Difference formula for tangent expresses tan(A−B) in terms of tanA and tanB: tan(A−B)=1+tanAtanBtanA−tanB
These formulas are essential for angle addition and angle subtraction in trigonometric functions
Cofunction identities in formulas
Cofunction identities relate trigonometric functions of complementary angles (2π−θ)
Sine and cosine are cofunctions: sin(2π−θ)=cosθ and cos(2π−θ)=sinθ
Tangent and cotangent are cofunctions: tan(2π−θ)=cotθ and cot(2π−θ)=tanθ
Secant and cosecant are cofunctions: sec(2π−θ)=cscθ and csc(2π−θ)=secθ
Substitute cofunction identities into sum and difference formulas when angles are in the form 2π−θ to simplify expressions
Verification of trigonometric identities
Simplify the left-hand side (LHS) and right-hand side (RHS) of the identity separately by applying sum and difference formulas and utilizing cofunction identities when necessary
Compare the simplified LHS and RHS to verify their equality
If the simplified LHS and RHS are identical, the identity is verified
Solving equations with identities
Rewrite the trigonometric equation using sum and difference formulas and substitute cofunction identities if required
Solve the resulting equation for the unknown variable using algebraic techniques (factoring, quadratic formula, or linear equation solving)
Determine the solutions within the given domain, considering the period of the trigonometric functions involved (2π for sine and cosine, π for tangent and cotangent)
Simplification of complex expressions
Identify opportunities to apply sum and difference formulas within the trigonometric expression, particularly when sums or differences of angles appear within trigonometric functions
Rewrite the expression using the appropriate sum and difference formulas and utilize cofunction identities when necessary
Simplify the resulting expression using algebraic techniques (combining like terms, factoring, or expanding)
Evaluate the simplified expression for given angle values, if required
Relationship to periodic functions and the unit circle
Sum and difference identities are derived from the properties of periodic functions and their behavior on the unit circle
These identities help in understanding how trigonometric functions combine and interact, which is crucial for analyzing complex periodic phenomena
The unit circle provides a geometric interpretation of these identities, allowing for visual representation of angle addition and subtraction
Applying Sum and Difference Identities
Solving equations with identities
Example: Solve sin(2x)+sinx=1 for 0≤x≤2π
Rewrite sin(2x) using the double angle formula: sin(2x)=2sinxcosx
Substitute: 2sinxcosx+sinx=1
Factor out sinx: sinx(2cosx+1)=1
Solve sinx=1 and 2cosx+1=1 separately
sinx=1 yields x=2π
2cosx+1=1 yields cosx=0, so x=2π,23π
Solutions within the given domain: x=2π
Simplification of complex expressions
Example: Simplify tan(x+4π)−tan(x−4π)
Apply the sum and difference formulas for tangent: