TRIGONOMETRÍA: Cos (a+b) Demostración de la fórmula Academia DIEGO

Summary

In this tutorial, the instructor demonstrates the proof of the cosine of the sum of two angles formula. Using two adjacent right triangles and applying trigonometric relationships like cosine and sine, the instructor explains how the formula is derived step-by-step. Through the use of similar triangles and substitution, the formula for cos(a + b) = cos(a)cos(b) - sin(a)sin(b) is proven. The video provides an easy-to-follow breakdown of this important trigonometric identity, making it accessible for students learning about angle sum formulas in trigonometry.

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Q & A

What is the main topic of the video tutorial?

The main topic is the derivation of the cosine addition formula, specifically for the sum of two angles, using geometric principles and right triangles.

What is the starting point for the proof of the cosine addition formula?

The starting point is a diagram consisting of two adjacent right triangles sharing a common side, which are used to illustrate the relationship between angles and their corresponding trigonometric functions.

Why are the two right triangles considered similar in the proof?

The two triangles are considered similar because they share a common angle and have two sides that are parallel to each other, which satisfies the condition of similarity for triangles.

What does the cosine of an angle represent in this context?

In this context, the cosine of an angle represents the ratio of the adjacent side (cateto contiguo) to the hypotenuse of the right triangle.

How is the segment 'ave' defined in the proof?

The segment 'ave' is defined as the difference between the total length of one side (AC) and the segment BC, which is part of the right triangle, corresponding to the cosine of the angle A.

How is the segment BC related to the sine of the angle A?

The segment BC is equivalent to the product of the sine of angle A and the length of the hypotenuse, and it can be expressed in terms of the sine function.

What is the role of the trigonometric functions sine and cosine in the derivation?

Sine and cosine functions are used to express the lengths of various segments in terms of the angles of the right triangles, which are then substituted into the equation to derive the cosine addition formula.

Why is it important to define the lengths of specific segments like 'ac' and 'bc'?

Defining the lengths of these segments allows for the substitution of known values into the equation, helping to derive the final formula for the sum of two angles.

What is the final formula derived in the video tutorial?

The final formula derived is the cosine of the sum of two angles: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).

How does the video encourage further learning on the topic?

The video encourages further learning by suggesting that viewers can explore additional tutorials on other trigonometric formulas, such as those for the sine and tangent of angle sums, available on the channel.