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Bayesian linear regression
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Proofs of trigonometric identities are used to show relations between trigonometric functions. This article will list trigonometric identities and prove them.
Referring to the diagram at the right, the six trigonometric functions of θ are:
Ratio identities
The following identities are trivial algebraic consequences of these definitions and the division identity.
c is whatever value (not necessarily trigonometric), only to understand the simple demonstrations above.
That is because not appear in the graph.
Complementary angle identities
Two angles whose sum is π/2 radians (90 degrees) are complementary. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining:
But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields
It follows that
.
Expanding the square and simplifying on the left hand side of the equation gives
.
Because the imaginary and real parts have to be the same, we are left with the original identities
,
and also
.
Half-angle identities
The two identities giving the alternative forms for cos 2θ lead to the following equations:
The sign of the square root needs to be chosen properly—note that if 2π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore the correct sign to use depends on the value of θ.
For the tan function, the equation is:
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:
Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
Replace each of , , and with their complementary angles, so cotangents turn into tangents and vice-versa.
Now if
then
so the result follows from the triple tangent identity.
Prosthaphaeresis identities
Inequalities
The figure at the right shows a sector of a circle with radius 1. The sector is θ/(2π) of the whole circle, so its area is θ/2.
The area of triangle OAD is AB/2, or sinθ/2. The area of triangle OCD is CD/2, or tanθ/2.
Since triangle OAD lies completely inside the sector, which in turn lies completely inside triangle OCD, we have
This geometric argument applies if 0<θ<π/2. It 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 sinθ≤1 (because of the Pythagorean identity), so sinθ<θ. So we have
For negative values of θ we have, by symmetry of the sine function
Hence
Identities involving calculus
Preliminaries
These can be seen from looking at the diagrams.
Sine and angle ratio identity
Proof: From the previous inequalities, we have, for small angles
, so
, so
, or
, so
, but
, so
Cosine and angle ratio identity
Proof:
The limits of those three quantities are 1, 0, and 1/2, so the resultant limit is zero.
Cosine and square of angle ratio identity
Proof:
As in the preceding proof,
The limits of those three quantities are 1, 1, and 1/2, so the resultant limit is 1/2.
Proof of Compositions of trig and inverse trig functions
All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function
Proof:
We start from
Then we divide this equation by
Then use the substitution , also use the Pythagorean trigonometric identity:
↑
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