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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.
Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.
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. Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas ...
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
Xuan tu or Hsuan thu (simplified Chinese: 弦图; traditional Chinese: 絃圖; pinyin: xuántú; Wade–Giles: hsüan 2 tʻu 2) is a diagram given in the ancient Chinese astronomical and mathematical text Zhoubi Suanjing indicating a proof of the Pythagorean theorem. [1] Zhoubi Suanjing is one of the oldest Chinese texts on mathematics. The ...
Fig. 6 – A short proof using trigonometry for the case of an acute angle. Using more trigonometry, the law of cosines can be deduced by using the Pythagorean theorem only once. In fact, by using the right triangle on the left hand side of Fig. 6 it can be shown that:
English: Construction by New Orleans students Calcea Johnson and Ne’Kiya Jackson that yields a trigonometric proof of the Pythagorean theorem for a<b. [1] The proof is valid for almost all right triangles, except for the case when a=b, i.e. Isosceles Right Triangle.
The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa).
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