Trigonometric Calculus - Right Triangle
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FAQ/ Information
Trigonometry (trigone: triangle and metry: measures) is the branch of Mathematics that studies the proportion, fixing, between the lengths of the sides of a right triangle, to the various values of one of its acute angles. (Among these angles, the 30°, 45°, and 60° angles are called remarkable angles.) The proportions between the 3 sides of the right triangles are called sine, cosine, tangent, cotangent, among several others, depending on the sides considered in the proportion.
Source: pt.wikipedia.org
Source: pt.wikipedia.org
The Trigonometric Circle is a feature created to facilitate the visualization of these proportions between the sides of the right triangles. It consists of an oriented circle of unit radius, centered on the origin of the 2 axes of an orthogonal Cartesian plane, that is, a plane defined by two lines perpendicular to each other, both with a value of 0 (zero) at the point where they are cut. There are two directions of marking the arcs in the circle: the positive direction, called counterclockwise, which occurs from the origin of the arcs to the terminal side of the angle corresponding to the arc; and the negative direction, or clockwise, which occurs in the opposite direction to the previous one.
Source: pt.wikipedia.org
Source: pt.wikipedia.org
Sine
Given a right triangle, the sine of one of its 2 acute angles is the ratio between the length of the side opposite this angle and the length of the hypotenuse, calculated, as any ratio, by dividing one value by the other, the reference of the ratio.
In the trigonometric circle, the sine of any angle can be visualized in the projection of its radius (by definition equal to 1) onto the vertical axis.
Since sine is this projection and the radius of the trigonometric circle is equal to 1, it follows that
, That is, the image of the sine is the closed interval 
Cosine
Given a right triangle, the cosine of one of its 2 acute angles is the ratio between the length of the side adjacent to this angle and the length of the hypotenuse, calculated, like all ratios, by dividing one value by another, the reference of the ratio.
In the trigonometric circle, the cosine of any angle can be visualized in the projection of its radius (by definition equal to 1) onto the horizontal axis.
Since the cosine is this projection, and the radius of the trigonometric circle is equal to 1, it follows that,
, That is, the image of the cosine is the closed interval
Tangent
Given a right triangle, the tangent of one of its 2 acute angles is the ratio between the length of the side opposite this angle and the length of the side adjacent to it, calculated, as any ratio, by dividing one value by another, the reference of the ratio.
In the trigonometric circle, the tangent value of any angle can be visualized on the vertical line that tangents this circle at the point where it cuts the horizontal axis on the right side. On this line tangent to the trigonometric circle, the value of the trigonometric tangent of any angle is represented by the segment that runs from the point where it cuts the horizontal axis to the point where it cuts the line containing the radius of the trigonometric circle to the angle considered. To evaluate this value, it should be compared with the radius of the trigonometric circle, which by definition is equal to 1, preferably when this radius is over the upper part of the vertical orthogonal axis. Note that while sine and cosine are always smaller than the radius of the trigonometric circle and therefore smaller than 1, the trigonometric tangent can be both smaller and greater than 1.
Source: pt.wikipedia.org
Given a right triangle, the sine of one of its 2 acute angles is the ratio between the length of the side opposite this angle and the length of the hypotenuse, calculated, as any ratio, by dividing one value by the other, the reference of the ratio.
In the trigonometric circle, the sine of any angle can be visualized in the projection of its radius (by definition equal to 1) onto the vertical axis.
Since sine is this projection and the radius of the trigonometric circle is equal to 1, it follows that
, That is, the image of the sine is the closed interval Cosine
Given a right triangle, the cosine of one of its 2 acute angles is the ratio between the length of the side adjacent to this angle and the length of the hypotenuse, calculated, like all ratios, by dividing one value by another, the reference of the ratio.
In the trigonometric circle, the cosine of any angle can be visualized in the projection of its radius (by definition equal to 1) onto the horizontal axis.
Since the cosine is this projection, and the radius of the trigonometric circle is equal to 1, it follows that,
, That is, the image of the cosine is the closed interval Tangent
Given a right triangle, the tangent of one of its 2 acute angles is the ratio between the length of the side opposite this angle and the length of the side adjacent to it, calculated, as any ratio, by dividing one value by another, the reference of the ratio.
In the trigonometric circle, the tangent value of any angle can be visualized on the vertical line that tangents this circle at the point where it cuts the horizontal axis on the right side. On this line tangent to the trigonometric circle, the value of the trigonometric tangent of any angle is represented by the segment that runs from the point where it cuts the horizontal axis to the point where it cuts the line containing the radius of the trigonometric circle to the angle considered. To evaluate this value, it should be compared with the radius of the trigonometric circle, which by definition is equal to 1, preferably when this radius is over the upper part of the vertical orthogonal axis. Note that while sine and cosine are always smaller than the radius of the trigonometric circle and therefore smaller than 1, the trigonometric tangent can be both smaller and greater than 1.
Source: pt.wikipedia.org
Pythagorean's theorem states that 'The sum of the square of the measurements of the legs (sides that form the angle of 90°, in this case c and b) is equal to the square of the measure of the hypotenuse (side opposite the angle of 90°, or a)'. So: a² = b² + c². A corollary of this theorem is that if the two sides are of the same size, the hypotenuse is worth the product of the side by the square root of 2.
Source: pt.wikipedia.org
Source: pt.wikipedia.org
There are several applications of trigonometry and trigonometric functions. For example, the triangulation technique is used in astronomy to estimate the distance of nearby stars; in geography to estimate distances between chevrons, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions, which describe sound and light waves.
Fields that make use of trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and therefore navigation (in oceans, on airplanes, and in space), music theory, acoustics, optics, market analysis, electronics, probability theory, statistics, biology, medical equipment (e.g., Computed Tomography and Ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many of the physical sciences, soils (inspection and geodesy), architecture, phonetics, economics, engineering, computer graphics, cartography, crystallography and game development.
Source: pt.wikipedia.org
Fields that make use of trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and therefore navigation (in oceans, on airplanes, and in space), music theory, acoustics, optics, market analysis, electronics, probability theory, statistics, biology, medical equipment (e.g., Computed Tomography and Ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many of the physical sciences, soils (inspection and geodesy), architecture, phonetics, economics, engineering, computer graphics, cartography, crystallography and game development.
Source: pt.wikipedia.org
A right triangle, in geometry, is a triangle in which one of the angles is straight (that is, an angle of 90 degrees). The relationship between the sides and angles of a right triangle is the basis of trigonometry.
The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called the corners. The A side can be identified as the side adjacent to the angle
And opposite the angle 
, While side b is the side adjacent to the angle And opposite the angle .
If the lengths of the three sides of a right triangle are integer, the triangle is considered a Pythagorean triangle and its lateral lengths are collectively known as a Pythagorean triple.
Source: pt.wikipedia.org
The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called the corners. The A side can be identified as the side adjacent to the angle
And opposite the angle , While side b is the side adjacent to the angle And opposite the angle .If the lengths of the three sides of a right triangle are integer, the triangle is considered a Pythagorean triangle and its lateral lengths are collectively known as a Pythagorean triple.
Source: pt.wikipedia.org