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  2. List of trigonometric identities - Wikipedia

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

    The triangle shaded blue illustrates the identity + ⁡ = ⁡, and the red triangle shows that ⁡ + = ⁡. The basic relationship between the sine and cosine is given by the Pythagorean identity: sin 2 ⁡ θ + cos 2 ⁡ θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}

  3. Pythagorean trigonometric identity - Wikipedia

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

    In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity ...

  4. Proofs of trigonometric identities - Wikipedia

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

    The figure at the right shows a sector of a circle with radius 1. The sector is θ/(2 π) of the whole circle, so its area is θ/2. We assume here that θ < π /2. = = = ⁡ = ⁡ The area of triangle OAD is AB/2, or sin(θ)/2.

  5. Mnemonics in trigonometry - Wikipedia

    en.wikipedia.org/wiki/Mnemonics_in_trigonometry

    The sum of the squares of the two items at the top of a triangle equals the square of the item at the bottom. These are the trigonometric Pythagorean identities : sin 2 ⁡ A + cos 2 ⁡ A = 1 2 = 1 {\displaystyle \sin ^{2}A+\cos ^{2}A=1^{2}=1\ }

  6. Small-angle approximation - Wikipedia

    en.wikipedia.org/wiki/Small-angle_approximation

    The quantity 206 265 ″ is approximately equal to the number of arcseconds in a circle (1 296 000 ″), divided by 2π, or, the number of arcseconds in 1 radian. The exact formula is = ⁡ (″) and the above approximation follows when tan X is replaced by X.

  7. Sine and cosine - Wikipedia

    en.wikipedia.org/wiki/Sine_and_cosine

    The fixed point iteration x n+1 = cos(x n) with initial value x 0 = −1 converges to the Dottie number. Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin ⁡ ( 0 ) = 0 {\displaystyle \sin(0)=0} .

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  9. Area of a triangle - Wikipedia

    en.wikipedia.org/wiki/Area_of_a_triangle

    Taking L to be the x-axis, the line integral between consecutive vertices (x i,y i) and (x i+1,y i+1) is given by the base times the mean height, namely (x i+1 − x i)(y i + y i+1)/2. The sign of the area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal.