### TRIGONOMETRY ASSIGNMENT HELP

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#### What Is Trigonometry?

Trigonometry is the study of rotation; in particular, things like “If I rotate this point by this angle around that point, where will it end up?”. It’s called “trigonometry” because, among other things, it can be used to analyze triangles (and if one breaks everything down into coordinates, it can alternatively be thought of as the analysis of right triangles in terms of their angles and leg lengths). Another way of looking at it is via so-called “complex numbers” (which formalize the arithmetic of rotation).

Trigonometry is of Two types. Analytical and Synthetic

Analytical Trigonometry :- Analytic Trigonometry is the fundamental concepts and underlying principle of analytic geometry.

Synthetic Trigonometry :- This is the art of defining and generating multiple shapes keeping the Theta(Angle) taken as a parameter and real number field is not used. Obviously, Line Segments don’t follow normal Algebraic Structures and it is kind of non-commutative and nonassociative algebra but is combinatorial, recursive and fractal in nature.Trigonometry relates to right angle triangles. If x and y are acute angles, obviously the 3rd angle remaining is a right angle.i.e. 90° . and the sum of x+y= 90°. because sum of all angles of a triangle is always 180°. Now tan(x+y)= tan90° sin x+y = 90°. Now, sin90° = 1 & cos90°=0. Therefore tan90° = sin90°/cos90° = 1/0 which is not defined. Hence, tan(x+y) is not defined if sin square x + sin square y = 1.

Right Triangle :- A right triangle is a triangle which has a right angle as one of its three angles, and it is the largest angle of the three, while the other two are acute angles, i.e., angles whose measure is less than 90 degrees but whose measure is greater than zero degrees. A right angle is an angle whose measure is 90 degrees.

Hypotenuse :- The hypotenuse is the longest side of a right triangle. It is opposite the right angle.The square of its length is equal to the sum of the squares of the legs. Legs are opposite the acute angles.The altitude from the right angle cuts the hypotenuse into two pieces whose product is equal to the square of the altitude.The hypotenuse of a triangle inscribed in a circle goes through the center. Therefore the hypotenuse is a diameter of the circle.

Trigonometric Functions :-

• Sin x = opposite/ hypotenuse
• Cosec x = hypotenuse/ opposite

Trigonometric Properties :-

• Cot x = 1/tan x
• Cosec x = 1/ sin x
• Sec x = 1/ cos x
• Tan x = sin x/cos x
• Cot x = cos x/ sin x
• Tan x = sec x /cosec x
• Cot x = cosec x / sec x
• Sin(-x) = – sin x
• Cos(-x) = cos x
• Tan (-x) = -tan x
• Cosec(-x) = -cosec x
• Sec(-x) = sec x
• Cot(-x) = -cot x

Ptolemy’s Theorem :- The Ptolemy’s theorem relies on the Pythagoras theorem. Its response is laid by the theorem. This concept serves the cornerstones and the preparation of Trigonometry, which develops and establishes the connection between angles and sides of a triangle.According to the theorem, A is the angle of this triangle. While Cos A is the span sin A is the duration of the triangle opposite to the hypotenuse.

According to the theorem, saying and the deduction is made :-

Sin (A+ B) = sin (A) cos (B) + cos (A) sin (B)

Sin (A- B) =  sin (A) cos (B) – cos (A) sin (B)