A Synopsis on Trigonometric Ratios of Multiple and Submultiple Angles

The branch of mathematics known as trigonometry is concerned with angles and the many different ways in which angle measurement can be accomplished. In order to arrive at results that are consistent, the application of trigonometric functions, such as trigonometric ratios, compound  angles, and multiple angles, is required. An example of an angle is a ray that is rotated around its starting point to create a new angle. The side of the object that is facing away from the point where the rotation starts is referred to as the initial side, and the side of the object that is facing toward the point where it stops is referred to as the terminal side. The angle that is formed as a result of turning a ray in the anticlockwise direction produces a positive outcome, whereas the angle that is produced when the ray is rotated in the clockwise direction produces a negative result. When measuring angles, degrees and radians can both be used as appropriate units.

Pythagorean Trigonometric Ratios Identities 

The Pythagorean theorem functions as the point of departure for the purpose of determining the identities of the trigonometric ratios that make use of the Pythagorean system. The Pythagorean theorem, when applied to the triangle with a right angle, gives us the following results:

Opposite2 + Adjacent2 = Hypotenuse2

Using the Hypotenuse2 as a divider between the two sides

Hypotenuse2/Hypotenuse2 equals the combination of the opposite^2/Hypotenuse^2 and adjacent2/Hypotenuse2. 

sin2θ + cos2θ = 1

sin2θ + cos2θ = 1

1 + tan2θ = sec2θ

1 + cot2θ = cosec2θ

Submultiple Angles

A Compilation of Ratios Used in Trigonometry

• sin A/2 + cos A/2 = √(1 + sin A)
• sin A/2 – cos A/2 = √(1 – sin A)
• tan A/2 = ±√(1 – cos A)/(1 + cos A)
• sinA = 2sinA/2cosA/2
• cos A = cos²A/2 – sin²A/2
• tan²A/2 = 1-cosA1+cosA
• sinA =2tanA/21+tan²A/2
• sinA= 3sinA/3-4sin²A/3
• cosA =  4cos³A/3-3cosA/3
• cosA = 1-tan²A/2/1+tan²A/2
• sin 2A = 2 sin A cos A = 2 tan A/(1+tan2A)
• cos 2A = cos2A – sin2A  = 2cos2A-1 = 1-2 sin2A
• tan 2A = 2 tan A/(1-tan2A)
• tan 3A = (3 tan A- tan3A)/(1-3tan2A)

Compound Angles

The angle that is the result of adding up the measures of two or more separate angles is referred to as a compound angle. The trigonometric identities that relate to compound angles can be used to depict those compound angles. It is feasible to perform the fundamental arithmetic operations of computing the sum and difference of functions by utilising the concept of compound angles.

List of Trigonometric Ratios of Compound Angles

These are the formulas that can be used to calculate the trigonometric ratios of compound angles:

• sin (A + B) = sin A cos B + cos A sin B
• cos (A + B) = cos A cos B – sin A cos B
• cos (A – B) = cos A cos B + sin A cos B
• tan (A + B) = [tan A + tanB] / [1 – tan A tan B]
• tan (A – B) = [tan A – tan B] / [1 + tan A tan B]
• sin (A + B) sin (A – B) = sin^2 A – sin^2 B = cos^2 B – cos^2 A
• cos (A + B) cos (A – B) = cos^2 A – sin^2 A – sin^2 B = cos^2 B – sin^2 A

•  2 sin A cos B = sin (A+B) + sin (A-B)
•  2 cos A cos B = cos (A+B) + cos (A-B)
•  2 cos A sin B = sin (A+B) – sin (A-B)
•  2 sin A sin B = cos (A-B) – cos (A+B)

Sum, Difference, Product Trigonometric Ratios Identities

• cos (A + B) = cos A cos B – sin A sin B
• cos (A – B) = cos A cos B + sin A sin B
• tan (A + B) = (tan A + tan B)/ (1 – tan A tan B)
• tan (A – B) = (tan A – tan B)/ (1 + tan A tan B)
• cot (A + B) = (cot A cot B – 1)/(cot B – cot A)
• cot (A – B) = (cot A cot B + 1)/(cot B – cot A)
• 2 sin A⋅cos B = sin(A + B) + sin(A – B)
• 2 cos A⋅cos B = cos(A + B) + cos(A – B)
• 2 sin A⋅sin B = cos(A – B) – cos(A + B)

Conclusion

Mathematically speaking, trigonometry is the study of the relationships between the lengths of the sides and the angles of right triangles (triangles with one angle that is exactly 90 degrees). When describing the locations of the three sides that make up any triangle, the phrases opposite and adjacent are used to indicate the locations of the sides that are adjacent to each other. The use of trigonometry and its functions can be found in an unusually wide variety of contexts, each of which calls for a unique set of skills. For instance, it is utilised in the field of geography for the purpose of determining the distance between various points of interest, in the field of astronomy for the purpose of determining the distance between relatively close stars, and in the field of satellite navigation for the purpose of determining the distance between various satellites.