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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.
Nov 14, 2024 · (Beyer 1987, p. 140). Formulas of these types can also be given analytically as
Sep 27, 2024 · Range of sin function = [-1, 1] Period of Sine Function. Since Sine function is a periodic function, we can define the time period after which the values of Sine function begin to repeat. The period of Sine function is 2π and can be written as: sin (2nπ + x) = sin x (for all n ∈ integer) Period of sin function = 2π. For instance, sin(2π) = 0.
sin ^2 (x) + cos ^2 (x) = 1 . tan ^2 (x) + 1 = sec ^2 (x) . cot ^2 (x) + 1 = csc ^2 (x) . sin(x y) = sin x cos y cos x sin y
- Right Triangle
- Sine, Cosine and Tangent
- Cosecant, Secant and Cotangent
- Pythagoras Theorem
The Trigonometric Identities are equations that are true for Right Angled Triangles. (If it is not a Right Angled Triangle go to the Triangle Identitiespage.) Each side of a right trianglehas a name:
The three main functions in trigonometry are Sine, Cosine and Tangent. They are just the length of one side divided by another For a right triangle with an angle θ: For a given angle θ each ratio stays the same no matter how big or small the triangle is When we divide Sine by Cosine we get: sin(θ)cos(θ) = Opposite/HypotenuseAdjacent/Hypotenuse = Op...
We can also divide "the other way around" (such as Adjacent/Opposite instead of Opposite/Adjacent): Because of all that we can say: And the other way around: And we also have:
For the next trigonometric identities we start with Pythagoras' Theorem: Dividing through by c2gives a2 c2 + b2 c2 = c2 c2 This can be simplified to: (a c )2 + (b c )2= 1 Now, a/c is Opposite / Hypotenuse, which is sin(θ) And b/c is Adjacent / Hypotenuse, which is cos(θ) So (a/c)2 + (b/c)2= 1 can also be written: Related identities include:
To solve a trigonometric simplify the equation using trigonometric identities. Then, write the equation in a standard form, and isolate the variable using algebraic manipulation to solve for the variable.
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The sine function sin takes angle θ and gives the ratio opposite hypotenuse . The inverse sine function sin-1 takes the ratio oppositehypotenuse and gives angle θ. And cosine and tangent follow a similar idea.
