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Degrees sin cos tan cot sec csc ... the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2. [5] ... The value of may be derived ...
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.
The parabola which also passes through (30, 1/2) (which is the point corresponding to the value sin(30°) = 1/2) is = (). These expressions suggest a varying denominator which takes the value 90 × 90 when x = 90 and the value 2 × 30 × 150 when x = 30.
For real number x, the notation sin x, cos x, etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.).
[1] [2] One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision. There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions.
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} .
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 ...
In radians, one would require that 0° ≤ x ≤ π/2, that x/π be rational, and that sin(x) be rational. The conclusion is then that the only such values are sin(0) = 0, sin(π/6) = 1/2, and sin(π/2) = 1. The theorem appears as Corollary 3.12 in Niven's book on irrational numbers. [2] The theorem extends to the other trigonometric functions ...