Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions that is true for every value of the variable for which both sides of the equation are defined. These are essential for simplifying expressions and solving trigonometric equations.

Fundamental Identities

Reciprocal and Quotient Identities

Reciprocal Identities
csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}
Quotient Identities
tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}

Even-Odd Identities

Even Functions (Symmetric about the y-axis)
cos⁡(−x)=cos⁡x,sec⁡(−x)=sec⁡x \cos(-x) = \cos x, \quad \sec(-x) = \sec x
Odd Functions (Symmetric about the origin)
sin⁡(−x)=−sin⁡x,csc⁡(−x)=−csc⁡x \sin(-x) = -\sin x, \quad \csc(-x) = -\csc x tan⁡(−x)=−tan⁡x,cot⁡(−x)=−cot⁡x \tan(-x) = -\tan x, \quad \cot(-x) = -\cot x

Co-function Identities

Co-function identities show the relationship between a trigonometric function and its co-function evaluated at the complementary angle.
  • sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta
  • cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta
  • tan⁡(π2−θ)=cot⁡θ\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta
  • csc⁡(π2−θ)=sec⁡θ\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta
  • sec⁡(π2−θ)=csc⁡θ\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta
  • cot⁡(π2−θ)=tan⁡θ\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta
Key Takeaways
  • Fundamental Identities: Reciprocal and Quotient identities are foundational for rewriting and simplifying complex trigonometric expressions.
  • Even-Odd Properties: Knowing which functions are even (Cosine, Secant) or odd helps handle negative angles easily.

Pythagorean Identities

Derived from the unit circle equation x2+y2=1x^2 + y^2 = 1:

Pythagorean Identities

Primary Identity:

Primary Pythagorean Identity

The fundamental relationship between sine and cosine based on the unit circle.

Variables

SymbolDescriptionUnit
θ\thetaAnglerad

Pythagorean Identities

Derived Identities: Divide by cos⁡2θ\cos^2 \theta:

Tangent and Secant Pythagorean Identity

Derived by dividing the primary identity by cosine squared.

Variables

SymbolDescriptionUnit
θ\thetaAnglerad

Pythagorean Identities

Divide by sin⁡2θ\sin^2 \theta:

Cotangent and Cosecant Pythagorean Identity

Derived by dividing the primary identity by sine squared.

Variables

SymbolDescriptionUnit
θ\thetaAnglerad
Key Takeaways
  • Pythagorean Substitutions: Extremely useful for proofs, often appearing as 1−sin⁡2θ=cos⁡2θ1 - \sin^2 \theta = \cos^2 \theta.

Angle Formulas

These formulas allow us to evaluate functions of sums or differences of angles.

Sum and Difference Formulas

  • sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B
  • cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
  • tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Double Angle Formulas

  • sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin \theta \cos \theta
  • cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta
  • tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta}

Triple Angle Formulas

  • sin⁡(3θ)=3sin⁡θ−4sin⁡3θ\sin(3\theta) = 3\sin \theta - 4\sin^3 \theta
  • cos⁡(3θ)=4cos⁡3θ−3cos⁡θ\cos(3\theta) = 4\cos^3 \theta - 3\cos \theta
  • tan⁡(3θ)=3tan⁡θ−tan⁡3θ1−3tan⁡2θ\tan(3\theta) = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}

Power-Reducing Formulas

These are derived by rearranging the double angle formulas for cosine. They are extremely important in calculus for integrating even powers of sine and cosine.
  • sin⁡2θ=1−cos⁡(2θ)2\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}
  • cos⁡2θ=1+cos⁡(2θ)2\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}
  • tan⁡2θ=1−cos⁡(2θ)1+cos⁡(2θ)\tan^2 \theta = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}

Half Angle Formulas

  • sin⁡(θ2)=±1−cos⁡θ2\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}
  • cos⁡(θ2)=±1+cos⁡θ2\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}
  • tan⁡(θ2)=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta}
Note: The sign (±\pm) depends on the quadrant of the half-angle θ/2\theta/2.
Key Takeaways
  • Double Angle: Useful for reducing powers (e.g., cos⁡2x=1+cos⁡2x2\cos^2 x = \frac{1+\cos 2x}{2}) and simplifying products of sine and cosine.
  • Sum/Difference: Let us evaluate trigonometric functions for non-standard angles (like 15∘15^\circ or 105∘105^\circ) by breaking them into combinations of standard angles.

Product-to-Sum and Sum-to-Product Formulas

These formulas are essential for transforming products of trigonometric functions into sums or differences, which is particularly useful in calculus for integration, and vice versa.

Product-to-Sum Identities

Converts products of sine and cosine into sums or differences.

Variables

SymbolDescriptionUnit
AAFirst anglerad
BBSecond anglerad

Sum-to-Product Identities

Converts sums or differences of sine and cosine into products.

Variables

SymbolDescriptionUnit
xxFirst anglerad
yySecond anglerad
Key Takeaways
  • Product-to-Sum: Crucial for integration techniques in calculus where products of sines and cosines are difficult to evaluate.
  • Sum-to-Product: Frequently used in physics and engineering to analyze wave interference and superposition.

Prosthaphaeresis

Before the invention of logarithms in the early 17th century, multiplying large numbers was extremely time-consuming and error-prone. Mathematicians and astronomers used a technique called Prosthaphaeresis (from Greek, meaning "addition and subtraction").
They used the Product-to-Sum formulas backward:
cos⁡Acos⁡B=12[cos⁡(A+B)+cos⁡(A−B)] \cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)]
To multiply two numbers, they would scale them to be between 0 and 1, look up the angles whose cosines matched those numbers in a trig table, add and subtract the angles, find the cosines of those new angles, and simply average them. This reduced multiplication to simple addition and subtraction!

Advanced Identities

Harmonic Addition Theorem

A linear combination of a sine and cosine wave of the same frequency can be written as a single sine or cosine wave with a phase shift. This is extremely useful in alternating current (AC) circuit analysis and wave mechanics.

Harmonic Addition Theorem (Sine Form)

Combines a sine and cosine wave into a single phase-shifted sine wave.

Variables

SymbolDescriptionUnit
A,BA, BAmplitudes of the original waves-
RRResultant amplitude (R = \sqrt{A^2 + B^2})-
α\alphaPhase shift angle (\tan \alpha = \frac{B}{A})rad
xxVariable angle or time-

Harmonic Addition Theorem

Where:
  • R=A2+B2R = \sqrt{A^2 + B^2} (the new amplitude)
  • tan⁡α=BA\tan \alpha = \frac{B}{A} (the phase shift angle, ensure α\alpha is in the correct quadrant based on the signs of AA and BB)
Alternatively, it can be expressed in terms of cosine:

Harmonic Addition Theorem (Cosine Form)

Combines a sine and cosine wave into a single phase-shifted cosine wave.

Variables

SymbolDescriptionUnit
RRResultant amplitude-
β\betaPhase shift angle (\tan \beta = \frac{A}{B})rad

Harmonic Addition Theorem

Where tan⁡β=AB\tan \beta = \frac{A}{B}.

Weierstrass Substitution (Half-Angle Tangent)

The tangent half-angle substitution, also known as the Weierstrass substitution, is a powerful technique for converting trigonometric expressions into rational algebraic expressions. It is primarily used to evaluate complex trigonometric integrals in calculus.
Let t=tan⁡(θ2)t = \tan\left(\frac{\theta}{2}\right). The fundamental trigonometric functions can then be expressed entirely in terms of tt:
sin⁡θ=2t1+t2 \sin \theta = \frac{2t}{1 + t^2} cos⁡θ=1−t21+t2 \cos \theta = \frac{1 - t^2}{1 + t^2} tan⁡θ=2t1−t2 \tan \theta = \frac{2t}{1 - t^2}
By substituting these algebraic equivalents, a difficult trigonometric problem is often reduced to a manageable algebraic one.

Euler's Formula and Identities

In advanced mathematics and engineering, memorizing all trigonometric identities is often unnecessary due to Euler's Formula:

Euler's Formula

The fundamental relationship between complex exponentials and trigonometric functions.

Variables

SymbolDescriptionUnit
eeEuler's number-
iiImaginary unit (\sqrt{-1})-
θ\thetaAnglerad

Euler's Formula and Identities

From this, sine and cosine can be expressed algebraically in terms of complex exponentials:
  • cos⁡θ=eiθ+e−iθ2\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}
  • sin⁡θ=eiθ−e−iθ2i\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}
By substituting these exponential definitions, any trigonometric identity (like Angle Sum or Double Angle) can be derived purely through basic algebraic expansion and exponent rules. This is the preferred method for dealing with trigonometry in electrical engineering and signal processing.

Proving Identities

To prove an identity, simplify one side (usually the more complex one) until it matches the other side. Do not treat it like an equation where you move terms across the equals sign.
Key Takeaways
  • Strategies for Proofs: Work on one side only. Rewrite everything in terms of sine and cosine. Combine fractions with a common denominator. Look for Pythagorean substitutions.
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