Inverse Multiple Angles

Inverse multiple angles are the inverse operation to find the sum of multiple angles. 

For example, if three angles are given as,

∠A, ∠B, ∠C

then: The measure of two adjacent (and supplementary) angles can be found using half their sum:

(∠A + ∠B)/2

This can be seen as the sum of two adjacent angles minus the third angle.

Using the sum of two angles: (∠A + ∠B)/2, the measure of two adjacent angles can be found.

To get a more general result, we may use the definition for angle “triangles,” i.e., one that is not supplementary.

Inverse Trigonometry

Using the inverse trigonometry formula, we can calculate all of Euler’s formulas, and so it can be used to find other trigonometric functions. This result will also be the same if we invert the angle using finding its opposite side.

We can calculate all formulas involving reciprocal, double, triple, and multiple angles using inverse multiple angle formulas.

However, this only gives us the full results for double angle and triple angle formulas.

Triangle Formula

Using the formula for single angle, we can find the formula for “inverse multiple angles”: 

Inverse Trigonometry: Using Euler’s formulas, we can get to the following definitions: 

Therefore, all formulas that involve multiple angles can be derived using trigonometric functions similarly as when calculating single angles.

Using the Sum of Three Angles

(∠A + ∠B + ∠C)

the measure of two adjacent angles can be found:

(∠A + ∠B + ∠C)/2

Proof: 

Using the definition for angle “triangles,” i.e., one that is not supplementary: 

(∠A + ∠B – ∠C)

Careful with the signs here! Angles “a” and “b” are adjacent (but not supplementary). Angles “c” and “d” are supplementary.

How to Find Inverse Multiple Angles? 

To find the “inverse multiple angles,” you need to invert all trigonometric functions involved. In this case, there are four functions, so it is easiest to use Euler’s formula.

In the example above, we found the “inverse multiple angles” (∠A + ∠B + ∠C) using Euler’s formula.

To get more generalized results and more general formulas for “inverse trigonometry,” inverse multiple angles may be constructed as: 

The inverse formula for single angles can be easily calculated from the formula for double angles. So with single angles, the formula for “inverse multiple angles” can be obtained from the formula for doubling.

Inverse multiple angles can also be understood as a line that has been rotated around itself and reflected at its feet. The real line has been shifted by 270 degrees along the bottom edge.

Inverse Contraction:

Similar to the above, we now consider the inverse form of addition when only two angles are given, i.e., ∠A and ∠B. 

We now write the sum of ∠A and ∠B to be:

(∠A + ∠B)/2

 (This is similar to subtracting two angles in a right triangle).

Inverse Multiple Angles

Let’s learn about the inverse multiple angles in trigonometry. 

In trigonometry, inverse multiple angles are given as 2θ, 3θ, 4θ and so forth. These are the reciprocal of trigonometric angles. 

Important inverse multiple angles is:

2θ = tan-¹ [2a/(1 – a²)] 

What is Multiple Angle Contraction?

Several of Euler’s formulas, such as the sine and cosine functions, require division by multiple angles. To do this, we must first calculate an inverse multiple angles. 

The sine function formula for multiple angle is as follows:

Sineθ =  Σk=0n cosk sinn-kθsin[½(n-k)] π

Here, n = 1,2,3……

Similarly for cosine function:

Cosθ =  Σk=0n cosk sinn-kθcos[½(n-k)] π

Here, n = 1,2,3……

Note: For these to work, you must have properly scaled the angle “integers,” i.e., the angle measure must not be greater than or equal to 180 degrees.

Using Inverse multiple angles, we can calculate the value of sine and cosine for multiple angles and triangles.

We can also use inverse multiple angles to find the reciprocal, double, triple, and multiple angles of a given angle.

Inverse Trigonometry

The following formulas for multiple angles can be derived from the formula for single angles. The simplest way is to multiply them by “the inverse single” (i.e., the number subtracted from the angle size). However, you may also follow all steps from the introduction to find how other formulas for this topic can be derived and how they work out.

Area of a Triangle

We can similarly calculate areas of triangles as we did with the addition of angles:

Equal Triangles:

If we prove that two angles are equivalent, we can easily prove that they form multiple angles. This is because the sum of two angles (i.e., their naive sum) can be calculated as an angle in these categories: supplementary, complementary, and equivalent.

Supplementary Triangles:

This is similar to the addition of angles in a right triangle, where the same angles are added:

If two angles have a common side (i.e., they share an edge), then they must be supplementary for their sum to be equal to their larger angle.

Complementary Triangles: 

If we assume that two complementary angles are supplementary, we can extend this theorem from 180 degrees to a sum of 360 degrees. 

Conclusion

The area of a triangle can be calculated similarly as we found for the addition of angles:

Similar to the example above, if we assume that two angles are equivalent, they form an angle in an equal triangle. This means that the sum of their measures is equal to the measure of their larger angle divided by the product of their measures. Therefore, their sum can be calculated as multiple angles.

It is valid for all real numbers and all right (i.e., acute). The theorem can be extended to acute angles if we replace the formulas.