Sum and Product Transformations

Trigonometry is a subject of mathematics that deals with angles and the many methods for measuring them. The use of trigonometric functions such as trigonometric ratios, compound angles, and numerous angles is essential in order to achieve consistent results. A ray that is rotated around its initial point to form a new angle is an example of an angle. The beginning side of the object is the side facing away from the point where the rotation begins, and the terminal side is the side facing toward the point where it ends. The angle generated by rotating a ray in the anticlockwise direction generates a positive result, whereas the angle formed by rotating the beam in the clockwise direction produces a negative result. When measuring angles, both degrees and radians are acceptable units.

Pythagorean Triogometric 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 opposite²/Hypotenuse² and adjacent²/Hypotenuse².

sin²θ + cos²θ = 1

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = cosec²θ

Expressing Products as Sums

Although we have already learnt a variety of formulas that are helpful for expanding or simplifying trigonometric expressions, there are instances when we might need to express the product of cosine and sine as a sum. We are able to make use of the product-to-sum formulas, which allow us to express the sums of the products of trigonometric functions. First, let’s look into the cosine identity, and then we can go on to the sine identity.

Expressing Products as Sums for Cosine

From the sum and difference identities for cosine, we can construct the formula for the product-to-sum relationship. When we add the two equations together, we obtain the following:

cosαcosβ + sinαsinβ = cos(α−β)

+ cosαcosβ − sinαsinβ = cos(α+β)

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2cosαcosβ = cos(α−β) + cos(α+β)

After that, we separate the product of cosines by dividing it by 2, as follows:

cosαcosβ = 1/2 [cos(α−β) + cos(α+β)]

Expressing the Product of Sine and Cosine as a Sum

In the following step, we will take the sum and difference formulae for sine and use them to create the product-to-sum formula for sine and cosine. When we add the IDs for the total and the difference, we get:

sin(α+β) = sinαcosβ + cosαsinβ

+ sin(α−β) = sinαcosβ − cosαsinβ

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sin(α+β) + sin(α−β) = 2sinαcosβ

After that, we divide that result by 2 to get the cosine and sine product, which is:

sinαcosβ=1/2[sin(α+β)+sin(α−β)]

Expressing Products of Sines in Terms of Cosine

The identities for the sum and difference of cosine can also be used to construct the expression for the product of sines expressed in terms of cosine. In this instance, we will begin by subtracting the two cosine formulas, which are as follows:

cos(α−β) = cosαcosβ + sinαsinβ

– cos(α+β) = −(cosαcosβ − sinαsinβ)

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cos(α−β) − cos(α+β) = 2sinαsinβ

After that, we separate the product of sines by dividing it by two:

sinαsinβ = 1/2[cos(α−β)−cos(α+β)]

Similarly, we may obtain alternative product-to-sum formulas by expressing the product of cosines in terms of sines or expressing the product of cosines in terms of sines.

Expressing Sums as Products

For certain issues, the approach that we have just taken needs to be flipped around. We are able to express sums of sine or cosine as products thanks to the formulas that convert sums to products. These equations are able to be derived from the product-to-sum identities that have been presented. By way of illustration, the sum-to-product identity for sine can be derived using just a few simple substitutions. Let (u+v)/2 = α and (u−v)/2 = β.

Then, 

α + β = {(u+v)/2} + {(u−v)/2}

= 2u/2 = u

α – β = {(u+v)/2} – {(u−v)/2}

= 2v/2 =v

Therefore, after substituting the equations we came up with for and in the formula for the product-to-sum total, we have

sinαcosβ = 1/2[sin(α + β) + sin(α – β)]

sin((u+v)/2) cos((u−v)/2) = 1/2 [sinu + sinv]

2sin((u+v)/2) cos( (u−v)/2) = sinu + sinv

replace (α + β) and (α – β) with the appropriate values.

The remaining sum-to-product identities are generated in the same manner as the first.

Transformations of Functions

Transformations change the features of a function while keeping the original ones.

A transformation alters a basic function by applying preset procedures to it. Depending on the type of transformation, this modification will cause the graph of the function to move, shift, or stretch. Translations, reflections, rotations, and scaling are the four primary forms of transformations.

Translations

A translation moves each point in the same direction by a given distance. The addition or deletion of a constant from a function causes the movement. Let f(x) = x³ as an example. x³ + 2 is one possible translation of f(x). This would be viewed as “the positive y direction translation of f(x) by two.”

Reflections

When you reflect a function, the graph becomes a mirror image of the original function. This is accomplished by changing the sign of the input to the function. Let f(x)=x 5 be the function in question. F(x)=x 5 would be the mirror image of this function across the yy-axis. As a result, we can argue that f(x) is a y-axis mirror of f(x).

Rotations

A rotation is a transformation in which an object is “spun” around a fixed point called the centre of rotation. Although the premise is simple, it has the most complicated mathematical method of the transformations mentioned. The following formulas are used:

x₁ = x₀cos – y₀sin

y₁ = x₀cos + y₀sin

Where x₁ and y₁ are the rotated function’s new expressions, x₀ and y₀ are the original expressions from the function being transformed, and theta is the angle at which the function is to be rotated. Allow y = x² to be an example. If we rotate this function 90 degrees, we get the following:

[xsin( π/2 )+ycos( π/2 )] = [xcos( π/2  )-ysin( π/2  )]²

Scaling

The graph of the function can undergo a transformation known as scaling, which alters either the size or the shape of the graph. It is important to keep in mind that none of the transformations that we have covered up to this point have been able to alter the dimensions and contours of a function; rather, they have only changed the graphical output from one set of points to another set of points. As an example, let f(x) = x³ . The conclusion that can be drawn from this is that 2f(x) = 2x³. Every point that was present on the graph of the initial function has been multiplied by two, which has resulted in the graph appearing to have physically grown in height.

CONCLUSION

The sum-to-product identities serve the same purpose as the product-to-sum identities. The sum or difference of sines and/or cosines in a product is rewritten using these identities. We would utilise the product-to-sum IDs to validate the identity if you wanted to be sure. Identities for sum-to-product.

The product of sine and cosine functions is expressed as a sum using the product to sum formulas. These are generated from trigonometry’s sum and difference formulas. When solving the integrals of trigonometric functions, these formulas come in handy.