Comprehensive Trigonometric Identities and Formulas

Right Triangle and Circular Function Definitions

• Right triangle definitions (for 0 < \theta < \frac{\pi}{2}):
  • \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}
  • \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
  • \tan\theta = \frac{\text{opposite}}{\text{adjacent}}
  • \csc\theta = \frac{\text{hypotenuse}}{\text{opposite}}
  • \sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}}
  • \cot\theta = \frac{\text{adjacent}}{\text{opposite}}
  • Note: These are defined where the hypotenuse is nonzero and the adjacent/opposite legs exist.
• Circular (unit-circle) definitions (0 is any angle):
  • For a general angle, if a point on the circle is (x, y) with radius r, then
  • On the unit circle (r = 1): $x = \cos\theta$, $y = \sin\theta$.
  • Reciprocal relationships: \csc\theta = \frac{1}{\sin\theta}, \sec\theta = \frac{1}{\cos\theta}, \cot\theta = \frac{\cos\theta}{\sin\theta}.
  • Domain considerations: \tan\theta and \cot\theta are undefined where \cos\theta = 0 or \sin\theta = 0 respectively.

Reciprocal Identities

• \csc\theta = \frac{1}{\sin\theta}
• \sec\theta = \frac{1}{\cos\theta}
• \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}
• Also include the primary (already listed) ratio forms:
  • \sin\theta = \frac{\text{opp}}{\text{hyp}}
  • \cos\theta = \frac{\text{adj}}{\text{hyp}}
  • \tan\theta = \frac{\text{opp}}{\text{adj}}

Double-Angle Formulas

• \sin 2u = 2 \sin u \cos u
  \sin 2u = 2 \sin u \cos u
• \cos 2u = \cos^2 u - \sin^2 u
  \cos 2u = \cos^2 u - \sin^2 u
• Alternative forms for cos 2u:
  • \cos 2u = 2 \cos^2 u - 1
  • \cos 2u = 1 - 2 \sin^2 u
    \cos 2u = 2 \cos^2 u - 1 = 1 - 2 \sin^2 u
• \tan 2u = \frac{2 \tan u}{1 - \tan^2 u}
  \tan 2u = \frac{2 \tan u}{1 - \tan^2 u}

Tangent and Cotangent Identities

• Fundamental ratio forms:
  • \tan x = \frac{\sin x}{\cos x}
  • \cot x = \frac{\cos x}{\sin x}
    \tan x = \frac{\sin x}{\cos x}, \; \cot x = \frac{\cos x}{\sin x}
• Note on domains: these are defined where the denominators are nonzero.

Power-Reducing (Power-Reducing) Formulas

• \sin^2 u = \frac{1 - \cos(2u)}{2}
  \sin^2 u = \frac{1 - \cos(2u)}{2}
• \cos^2 u = \frac{1 + \cos(2u)}{2}
  \cos^2 u = \frac{1 + \cos(2u)}{2}
• \tan^2 u = \frac{1 - \cos(2u)}{1 + \cos(2u)}
  \tan^2 u = \frac{1 - \cos(2u)}{1 + \cos(2u)}
• Related identities (often listed with Pythagorean set):
  • \tan^2 u = \sec^2 u - 1
  • 1 + \tan^2 u = \sec^2 u

Pythagorean Identities

• Fundamental:
  • \sin^2 x + \cos^2 x = 1
    \sin^2 x + \cos^2 x = 1
• Variants:
  • 1 + \tan^2 x = \sec^2 x
  • 1 + \cot^2 x = \csc^2 x
    1 + \tan^2 x = \sec^2 x, \; 1 + \cot^2 x = \csc^2 x

Cofunction Identities

• For cofunctions (using the relationship with \tfrac{\pi}{2} - x):
  • \sin\left(\tfrac{\pi}{2} - x\right) = \cos x
  • \cos\left(\tfrac{\pi}{2} - x\right) = \sin x
  • \tan\left(\tfrac{\pi}{2} - x\right) = \cot x
  • \cot\left(\tfrac{\pi}{2} - x\right) = \tan x
  • \csc\left(\tfrac{\pi}{2} - x\right) = \sec x
  • \sec\left(\tfrac{\pi}{2} - x\right) = \csc x

Reduction Formulas (Even/Odd)

• For negative angles:
  • \sin(-x) = -\sin x
  • \cos(-x) = \cos x
  • \tan(-x) = -\tan x
  • \csc(-x) = -\csc x
  • \sec(-x) = \sec x
  • \cot(-x) = -\cot x

Sum-to-Product Formulas

• Sum of sines:
  • \sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)
    \sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)
• Sum of cosines:
  • \cos u + \cos v = 2 \cos\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)
    \cos u + \cos v = 2 \cos\left(\frac{u+v}{2}\right) \cos\left(\frac{u-v}{2}\right)
• Difference forms (or differences of sines):
  • \sin u - \sin v = 2 \cos\left(\frac{u+v}{2}\right) \sin\left(\frac{u-v}{2}\right)
    \sin u - \sin v = 2 \cos\left(\frac{u+v}{2}\right) \sin\left(\frac{u-v}{2}\right)
• Product-to-Sum (converting products to sums):
  • \sin u \sin v = \tfrac{1}{2} [\cos(u - v) - \cos(u + v)]
  • \cos u \cos v = \tfrac{1}{2} [\cos(u - v) + \cos(u + v)]
  • \sin u \cos v = \tfrac{1}{2} [\sin(u + v) + \sin(u - v)]
  • \cos u \sin v = \tfrac{1}{2} [\sin(u + v) - \sin(u - v)]

Sum and Difference Formulas (Angle Addition and Subtraction)

• Sine:
  • \sin(u \pm v) = \sin u \cos v \pm \cos u \sin v
    \sin(u \pm v) = \sin u \cos v \pm \cos u \sin v
• Cosine:
  • \cos(u \pm v) = \cos u \cos v \mp \sin u \sin v
    \cos(u \pm v) = \cos u \cos v \mp \sin u \sin v

Additional Notes and Connections

• These identities connect to foundational principles: relationship between sides and angles in right triangles, unit circle definitions, and symmetry properties of trig functions.
• They enable simplification, solving equations, and proving more advanced theorems in mathematics, physics, engineering, signal processing, and computer science.
• Practical implications include domain considerations (where denominators vanish), periodicity, and using cofunctions for complementary angles.