SENO, COSSENO e TANGENTE NO TRIÂNGULO RETÂNGULO

Summary

In this engaging lesson, the instructor celebrates their birthday while teaching trigonometry, focusing on right-angled triangles. Key concepts such as sine, cosine, and tangent are introduced with practical applications, explaining how to identify the sides of a triangle based on the given angle. The lesson demonstrates how to solve for unknowns using trigonometric functions and the Pythagorean theorem. Real-world examples, such as calculating distances traveled by an airplane and the height of a building, are used to solidify the concepts. A helpful mnemonic for solving trigonometric problems is also shared to ensure students retain the knowledge.

Takeaways

• 😀 Sine, cosine, and tangent are essential concepts in trigonometry, especially in right triangles, and are used to relate the angles and sides of the triangle.
• 😀 In a right triangle, the hypotenuse is the side opposite the 90º angle, while the other two sides are the 'catetos' (adjacent and opposite).
• 😀 The sine function relates the ratio of the opposite side to the hypotenuse. It is written as sine(θ) = opposite / hypotenuse.
• 😀 The cosine function relates the ratio of the adjacent side to the hypotenuse. It is written as cosine(θ) = adjacent / hypotenuse.
• 😀 The tangent function relates the ratio of the opposite side to the adjacent side. It is written as tangent(θ) = opposite / adjacent.
• 😀 The acronym 'SOH-CAH-TOA' can help remember the relationships: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
• 😀 To solve problems, identify the given angles and sides, and choose the appropriate trigonometric function (sine, cosine, or tangent) to find unknowns.
• 😀 For practical problems, use the trig ratios with known values (angles and sides) and solve algebraically to find unknowns (e.g., using cross-multiplication).
• 😀 The script demonstrates solving problems by applying trigonometric functions like sine and cosine to calculate missing side lengths, such as x and y in the examples.
• 😀 A common problem involves an airplane maintaining a constant angle and reaching a given altitude, where trig functions help determine the distance traveled by the airplane.
• 😀 The script highlights practical tips and shortcuts, such as recognizing the standard 30°-60°-90° triangle, where the opposite sides have known ratios to the hypotenuse.

Q & A

What is the definition of a right triangle in trigonometry?

A right triangle is a triangle that has one angle measuring exactly 90 degrees (a right angle).

What are the three sides of a right triangle called?

The three sides of a right triangle are the hypotenuse, which is opposite the right angle; the opposite side, which is opposite to a given angle; and the adjacent side, which is next to the given angle.

What are the trigonometric functions sine, cosine, and tangent used for?

These functions relate the angles of a right triangle to the lengths of its sides. Sine (sin) is the ratio of the opposite side to the hypotenuse, cosine (cos) is the ratio of the adjacent side to the hypotenuse, and tangent (tan) is the ratio of the opposite side to the adjacent side.

What does the acronym 'SOH-CAH-TOA' represent in trigonometry?

'SOH-CAH-TOA' is a mnemonic to remember the definitions of sine, cosine, and tangent: SOH stands for 'Sine = Opposite / Hypotenuse', CAH stands for 'Cosine = Adjacent / Hypotenuse', and TOA stands for 'Tangent = Opposite / Adjacent'.

In the example with the angle of 30 degrees, how did the speaker find the value of x (the opposite side)?

The speaker used the sine function. By using the fact that sine of 30° is equal to 1/2, and knowing that sine of 30° = opposite / hypotenuse, they set up the equation x / 20 = 1/2, which simplifies to x = 10.

How was the value of y (the adjacent side) found in the example with the 30° angle?

The speaker used the cosine function. By knowing that cosine of 30° is √3 / 2, and that cosine of 30° = adjacent / hypotenuse, they set up the equation y / 20 = √3 / 2, which simplifies to y = 10√3.

What key formula does the speaker use to solve right triangle problems involving angles and sides?

The speaker uses the trigonometric functions sine, cosine, and tangent, which are derived from the ratios of the sides of a right triangle. These formulas help solve for unknown sides or angles based on known values.

In the problem involving the airplane, why does the speaker use sine to find the distance traveled by the airplane?

The speaker uses sine because they are working with the opposite side (the height of the airplane) and the hypotenuse (the path traveled by the airplane). The relationship between the height and hypotenuse is described by sine.

How do you calculate the height of the building in the second problem, using trigonometric functions?

The speaker uses the sine function, with an angle of 60° and a hypotenuse of 80 meters. By setting up the equation sine(60°) = x / 80, where x is the height of the building, they solve for x = 80√3 / 2, which simplifies to 40√3 meters.

What strategy does the speaker use when faced with a problem involving two unknowns in a triangle?

The speaker suggests using additional information, such as identifying isosceles triangles or applying known angle relationships, to simplify the problem. This often leads to easier solutions by eliminating unnecessary variables.