Interface:Trigonometry

Trigonometry concerns the sine function (and others which can be derived from it, such as cosine, tangent, etc). Here we concern ourselves only with the sine of a real number (not a complex number).

We build on propositional logic, first-order logic, set theory and Interface:Complex numbers (which we are just using for real numbers).

 param (CLASSICAL Interface:Classical_propositional_calculus ) param (FIRSTORDER Interface:First-order_logic_with_quantifiability (CLASSICAL) ) param (SETS Interface:Set_theory (CLASSICAL FIRSTORDER) ) param (COMPLEX Interface:Complex_numbers (CLASSICAL FIRSTORDER SETS) ) 

The sine and cosine functions
In the future, we might prefer to define sin as a function (that is, a set, as will be developed in Relations when that is more complete) from the reals to the reals, but for now we define it as a term.  var (object α β θ) term (object (sin object)) stmt (SineRealToReal  ((θ ∈ (ℝ)) → ((sin θ) ∈ (ℝ)))) 

We define cosine similarly.  term (object (cos object)) stmt (CosineRealToReal  ((θ ∈ (ℝ)) → ((cos θ) ∈ (ℝ)))) 

Circle constant
The circle constant  is the circumference of a circle divided by its radius.  term (object (τ)) 

Some authors instead use a value of, which they call.

Relationship between sine and cosine
The sine and the cosine at a given point, squared, add to one, which is the Pythagorean trigonometric identity. TODO: need to say θ is real.  stmt (TrigonometryPythagorean  ((((sin θ) · (sin θ)) + ((cos θ) · (cos θ))) = (1))) 

The sine and cosine are the same, shifted by a quarter turn.  stmt (SineShift  ((θ ∈ (ℝ)) → ((sin (θ + ((τ) / (4)))) = (cos θ)))) 

Periodicity
The sine and cosine are periodic with period  (we do not state, here, that   is the smallest value which is a period, although it is).  stmt (SinePeriod  ((θ ∈ (ℝ)) → ((sin (θ + (τ))) = (sin θ)))) 

Negation
There is a simple formula for the sine or cosine of a negated quantity.  stmt (SineNegation  ((θ ∈ (ℝ)) → ((sin (- θ)) = (- (sin θ))))) stmt (CosineNegation  ((θ ∈ (ℝ)) → ((cos (- θ)) = (cos θ)))) 

Addition
There are expressions for taking the sine or cosine of a sum or difference. TODO: need to say α and β are real.

 stmt (SineAddition  ((sin (α + β)) = (((sin α) · (cos β)) + ((cos α) · (sin β))))) stmt (CosineAddition  ((cos (α + β)) = (((cos α) · (cos β)) − ((cos α) · (sin β))))) stmt (SineSubtraction  ((sin (α − β)) = (((sin α) · (cos β)) − ((cos α) · (sin β))))) stmt (CosineSubtraction  ((cos (α − β)) = (((cos α) · (cos β)) + ((cos α) · (sin β))))) </jh>

Trigonometry functions at particular points
There are exact expressions for values of the trigonometric functions at many common points (for example, any multiple of  radians is possible, although we only provide some of the more common ones).

 stmt (Sin0  ((sin (0)) = (0))) stmt (Sin12  ((sin ((τ) / (12))) = ((1) / (2)))) stmt (Sin8  ((sin ((τ) / (8))) = ((√ (2)) / (2)))) stmt (Sin6  ((sin ((τ) / (6))) = ((√ (3)) / (2)))) stmt (Sin4  ((sin ((τ) / (4))) = (1))) stmt (Sin2  ((sin ((τ) / (2))) = (0))) stmt (Sin1  ((sin (τ)) = (0)))

stmt (Cos0  ((cos (0)) = (1))) stmt (Cos12  ((cos ((τ) / (12))) = ((√ (3)) / (2)))) stmt (Cos8  ((cos ((τ) / (8))) = ((√ (2)) / (2)))) stmt (Cos6  ((cos ((τ) / (6))) = ((1) / (2)))) stmt (Cos4  ((cos ((τ) / (4))) = (0))) stmt (Cos2  ((cos ((τ) / (2))) = (- (1)))) stmt (Cos1  ((cos (τ)) = (1))) </jh>

Degree measure
Although radian measure is far more convenient in most of mathematics, for convenience we also provide an expression which allows us to specify an angle in degrees. The trigonometric functions themselves are unchanged; they still take an argument in radians. But the degree expression provides a conversion, so that, for example, writing  is equivalent to writing.

 term (object (° object)) stmt (TurnDegree  ((τ) = ((360) °)))

stmt (Sin30  ((sin ((30) °)) = ((1) / (2)))) stmt (Sin45  ((sin ((45) °)) = ((√ (2)) / (2)))) stmt (Sin60  ((sin ((60) °)) = ((√ (3)) / (2)))) stmt (Sin90  ((sin ((90) °)) = (1))) stmt (Sin180  ((sin ((180) °)) = (0))) stmt (Sin360  ((sin ((360) °)) = (0)))

stmt (Cos30  ((cos ((30) °)) = ((√ (3)) / (2)))) stmt (Cos45  ((cos ((45) °)) = ((√ (2)) / (2)))) stmt (Cos60  ((cos ((60) °)) = ((1) / (2)))) stmt (Cos90  ((cos ((90) °)) = (0))) stmt (Cos180  ((cos ((180) °)) = (- (1)))) stmt (Cos360  ((cos ((360) °)) = (1))) </jh>