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Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
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.
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: (/) =
The starting corner equals the product of its two nearest neighbors. For example, sin A = cos A ⋅ tan A {\\displaystyle \\sin A=\\cos A\\cdot \\tan A} The sum of the squares of the two items at the top of a triangle equals the square of the item at the bottom.
Figure 7b cuts a hexagon in two different ways into smaller pieces, yielding a proof of the law of cosines in the case that the angle γ is obtuse. We have in pink, the areas a 2, b 2, and −2ab cos γ on the left and c 2 on the right; in blue, the triangle ABC twice, on the left, as well as on the right.
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[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.
As a special case, for C = π / 2 , then cos C = 0, and one obtains the spherical analogue of the Pythagorean theorem: cos c = cos a cos b {\displaystyle \cos c=\cos a\cos b\,} If the law of cosines is used to solve for c , the necessity of inverting the cosine magnifies rounding errors when c is small.