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Similar right triangles illustrating the tangent and secant trigonometric functions Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot 2 θ = csc 2 θ, and applied to the red triangle shows that 1 + tan 2 θ = sec 2 θ.
All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse.
The area of triangle OCD is CD/2, or tan(θ)/2. Since triangle OAD lies completely inside the sector, ... But sin θ ≤ 1 (because of the Pythagorean identity), ...
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is It is sin 2 x + cos 2 x = 1 {\displaystyle \sin ^{2}x+\cos ^{2}x=1} .
The following trigonometric identities are related to the Pythagorean theorem and hold for any value: [87] sin 2 A + cos 2 A = 1 {\displaystyle \sin ^{2}A+\cos ^{2}A=1\ } tan 2 A + 1 = sec 2 A {\displaystyle \tan ^{2}A+1=\sec ^{2}A\ }
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
The angle between the horizontal line and the shown diagonal is 1 / 2 (a + b). This is a geometric way to prove the particular tangent half-angle formula that says tan 1 / 2 (a + b) = (sin a + sin b) / (cos a + cos b). The formulae sin 1 / 2 (a + b) and cos 1 / 2 (a + b) are the ratios of the actual distances to ...
Then by the Pythagorean theorem, the length of the hypotenuse of such a triangle is . Scaling the triangle so that its hypotenuse has a length of 1 divides the lengths by 2 {\displaystyle {\sqrt {2}}} , giving the same value for the sine or cosine of 45° given above.
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