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  2. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides.

  3. Small-angle approximation - Wikipedia

    en.wikipedia.org/wiki/Small-angle_approximation

    Using the squeeze theorem, [4] we can prove that ⁡ =, which is a formal restatement of the approximation ⁡ for small values of θ. A more careful application of the squeeze theorem proves that lim θ → 0 tan ⁡ ( θ ) θ = 1 , {\displaystyle \lim _{\theta \to 0}{\frac {\tan(\theta )}{\theta }}=1,} from which we conclude that tan ⁡ ( θ ...

  4. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools.

  5. Pythagorean trigonometric identity - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_trigonometric...

    Similar right triangles illustrating the tangent and secant trigonometric functions Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ.

  6. Differentiation of trigonometric functions - Wikipedia

    en.wikipedia.org/wiki/Differentiation_of...

    All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation .

  7. Trump’s extreme dislike of FBI directors, explained - AOL

    www.aol.com/news/trump-extreme-dislike-fbi...

    President-elect Donald Trump is still putting together his Cabinet for his second term, and many other government positions will open up when he takes the oath of office in January.

  8. Tangent half-angle formula - Wikipedia

    en.wikipedia.org/wiki/Tangent_half-angle_formula

    The sides of this rhombus have length 1. The angle between the horizontal line and the shown diagonal is ⁠ 1 / 2 ⁠ (a + b).This is a geometric way to prove the particular tangent half-angle formula that says tan ⁠ 1 / 2 ⁠ (a + b) = (sin a + sin b) / (cos a + cos b).

  9. Law of tangents - Wikipedia

    en.wikipedia.org/wiki/Law_of_tangents

    To prove the law of tangents one can start with the law of sines: a sin ⁡ α = b sin ⁡ β = d , {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}=d,} where ⁠ d {\displaystyle d} ⁠ is the diameter of the circumcircle , so that ⁠ a = d sin ⁡ α {\displaystyle a=d\sin \alpha } ⁠ and ⁠ b = d sin ⁡ β {\displaystyle ...