If sinθ is the GM of sinΦ and cosΦ ,Prove cos2θ= 2 cos^2(π/4+Φ) I XI I Trigonometry

If sinθ is the GM of sinΦ and cosΦ ,Prove cos2θ= 2 cos^2(π/4+Φ) I XI I Trigonometry I CBSE I NCERT I ICSE

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. ... The word trigonometry is a 16th-century Latin derivative from the Greek words for triangle (trigōnon) and measure (metron).
Trigonometry, the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure. For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A, or sin A; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.
Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry is found all throughout geometry, as every straight-sided shape may be broken into as a collection of triangles. Further still, trigonometry has astoundingly intricate relationships to other branches of mathematics, in particular complex numbers, infinite series, logarithms and calculus.

The word trigonometry is a 16th-century Latin derivative from the Greek words for triangle (trigōnon) and measure (metron). Though the field emerged in Greece during the third century B.C., some of the most important contributions (such as the sine function) came from India in the fifth century A.D. Because early trigonometric works of Ancient Greece have been lost, it is not known whether Indian scholars developed trigonometry independently or after Greek influence.
Sine, cosine and tangent
Depending on what is known about various side lengths and angles of a right triangle, there are two other trigonometric functions that may be more useful: the “sine function” written as sin(x), and the “cosine function” written as cos(x). Before we explain those functions, some additional terminology is needed. Sides and angles that touch are described as adjacent. Every side has two adjacent angles. Sides and angles that don’t touch are described as opposite. For a right triangle, the side opposite to the right angle is called the hypotenuse (from Greek for “stretching under”). The two remaining sides are called legs.

In other words:

The tangent of angle A = the length of the opposite side divided by the length of the adjacent side
The sine of angle A = the length of the opposite side divided by the length of the hypotenuse
The cosine of angle A = the length of the adjacent side divided by the length of the hypotenuse
From our ship-mast example before, the relationship between an angle and its tangent can be determined from its graph, shown below. The graphs of sine and cosine are included as well.
Worth mentioning, though beyond the scope of this article, is that these functions relate to each other through a great variety of intricate equations known as identities, equations that are always true.

Each trigonometric function also has an inverse that can be used to find an angle from a ratio of sides. The inverses of sin(x), cos(x), and tan(x), are arcsin(x), arccos(x) and arctan(x), respectively.
Shapes other than right triangles
Trigonometry isn’t limited to just right triangles. It can be used with all triangles and all shapes with straight sides, which are treated as a collection of triangles. For any triangle, across the six measures of sides and angles, if at least three are known the other three can usually be determined. Of the six configurations of three known sides and angles, only two of these configurations can’t be used to determine everything about a triangle: three known angles (AAA), and a known angle adjacent and opposite to the known sides (ASS). Unknown side lengths and angles are determined using the following tools:

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