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  2. List of trigonometric identities - Wikipedia

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

    A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.

  3. Sine and cosine - Wikipedia

    en.wikipedia.org/wiki/Sine_and_cosine

    In mathematics, sine and cosine are trigonometric functions of an angle.The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that ...

  4. Law of sines - Wikipedia

    en.wikipedia.org/wiki/Law_of_sines

    In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, ⁡ = ⁡ = ⁡ =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.

  5. Exact trigonometric values - Wikipedia

    en.wikipedia.org/wiki/Exact_trigonometric_values

    If the denominator, b, is multiplied by additional factors of 2, the sine and cosine can be derived with the half-angle formulas. For example, 22.5° ( π /8 rad) is half of 45°, so its sine and cosine are: [ 11 ]

  6. Law of cosines - Wikipedia

    en.wikipedia.org/wiki/Law_of_cosines

    It is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for a in the quadratic equation a 2 − 2ab cos γ + b 2 − c 2 = 0. This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data.

  7. Proofs of trigonometric identities - Wikipedia

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

    This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...

  8. Pythagorean trigonometric identity - Wikipedia

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

    By the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Next we prove the identity in the range ⁠ π / 2 ⁠ < θ ≤ π. To do this we let t = θ − ⁠ π / 2 ⁠, t will now be in the range 0 < t ≤ π/2. We can then make use of squared versions of some basic shift identities ...

  9. Trigonometry - Wikipedia

    en.wikipedia.org/wiki/Trigonometry

    The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [32] With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. [33]