- Double‐Angle and Half‐Angle Identities
- Summary of trigonometric identities
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Trigonometry in the modern sense began with the Greeks. Hipparchus c.
Double‐Angle and Half‐Angle Identities
He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord that is, a straight line that connects two points on a curve or surface , as shown by the inscribed triangle A B C in the figure. To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends it—or, equivalently, the length of a chord as a function of the corresponding arc width.
This became the chief task of trigonometry for the next several centuries.
As an astronomer, Hipparchus was mainly interested in spherical triangles, such as the imaginary triangle formed by three stars on the celestial sphere , but he was also familiar with the basic formulas of plane trigonometry. The first major ancient work on trigonometry to reach Europe intact after the Dark Ages was the Almagest by Ptolemy c.
Summary of trigonometric identities
He lived in Alexandria , the intellectual centre of the Hellenistic world, but little else is known about him. In order to develop this world picture—the essence of which was a stationary Earth around which the Sun , Moon , and the five known planets move in circular orbits —Ptolemy had to use some elementary trigonometry.
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Identity - Plaid
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Exam MS-100: Microsoft 365 Identity and Services
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The above identities can be re-stated by squaring each side and doubling all of the angle measures.
The results are as follows:. You will be using all of these identities, or nearly so, for proving other trig identities and for solving trig equations.
However, if you're going on to study calculus, pay particular attention to the restated sine and cosine half-angle identities , because you'll be using them a lot in integral calculus. All right reserved. Web Design by. Skip to main content.