Trigonometry: Double/Triple Angle | Traditional Summary
Contextualization
Trigonometry is a fundamental area of mathematics that studies the relationships between the angles and sides of triangles. Within this field, the double and triple angle formulas are essential tools that allow for the simplification and resolution of complex problems involving trigonometric functions. For example, the double angle formula for sine, cosine, and tangent enables us to find relationships between the value of a trigonometric function at an angle and its double. These formulas are derived from basic trigonometric identities and have vast practical applications, ranging from solving mathematical problems to engineering and physics.
Additionally, the triple angle formulas further expand these relationships, allowing for precise calculations in triple angles. Understanding these formulas is crucial for developing advanced mathematical skills and applying them in various scientific and technological disciplines. For instance, in computer graphics, these formulas are used to calculate movements and positions of objects in a three-dimensional environment. In engineering, they help determine forces and stresses in complex structures like bridges and buildings. Thus, the study of double and triple angle formulas not only enriches mathematical knowledge but also opens doors to various practical applications in the real world.
Double Angle Formula for Sine
The double angle formula for sine is represented as sin(2x) = 2sin(x)cos(x). This formula is derived from basic trigonometric identities and allows us to express the sine of the double of an angle in terms of the sine and cosine of the original angle. The formula is useful for simplifying trigonometric expressions and solving problems where it is necessary to find the value of the sine of a double angle.
To understand how this formula is derived, let's consider the sum of angles. We know that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). If we substitute a for x and b for x, we get sin(x + x) = sin(x)cos(x) + cos(x)sin(x), which simplifies to sin(2x) = 2sin(x)cos(x). This derivation shows the fundamental relationship between the sine of an angle and its double.
The practical application of this formula can be observed in examples such as calculating sin(60º), where we know that 60º is double 30º. Using the formula, we can find sin(60º) = 2sin(30º)cos(30º), with known values of sin(30º) = 1/2 and cos(30º) = √3/2, resulting in sin(60º) = √3/2.
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Formula: sin(2x) = 2sin(x)cos(x)
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Derived from the sum of angles: sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
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Practical application: Calculation of sin(60º) using the double angle formula
Double Angle Formula for Cosine
The double angle formula for cosine is represented by cos(2x) = cos²(x) - sin²(x) and has variations such as cos(2x) = 2cos²(x) - 1 and cos(2x) = 1 - 2sin²(x). These formulas are derived from trigonometric identities and are useful for simplifying calculations involving the cosine of a double angle.
To derive the basic formula, we can use the fundamental identity of trigonometry, cos²(x) + sin²(x) = 1. From this identity, we can substitute cos²(x) and sin²(x) into the variations of the double angle formula. For example, replacing sin²(x) with 1 - cos²(x) in the formula cos(2x) = cos²(x) - sin²(x), we obtain cos(2x) = 2cos²(x) - 1.
The practical application of this formula can be seen in calculations such as finding the value of cos(22.5º). We know that 22.5º is half of 45º, and using the formula cos(2x) = 2cos²(x) - 1, we can solve for cos(x) when cos(45º) = √2/2, resulting in cos(22.5º) = √[(√2 + 2)/4].
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Formula: cos(2x) = cos²(x) - sin²(x)
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Variations: cos(2x) = 2cos²(x) - 1 and cos(2x) = 1 - 2sin²(x)
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Practical application: Calculation of cos(22.5º) using the double angle formula
Double Angle Formula for Tangent
The double angle formula for tangent is represented by tan(2x) = (2tan(x)) / (1 - tan²(x)). This formula is derived from basic trigonometric identities and is useful for simplifying calculations involving the tangent of a double angle.
To derive this formula, we can start with the tangent as the ratio between sine and cosine: tan(x) = sin(x)/cos(x). Using the double angle formulas for sine and cosine, we can express tan(2x) in terms of tan(x). Dividing the double angle formula for sine by the double angle formula for cosine, we obtain tan(2x) = (2tan(x)) / (1 - tan²(x)).
The practical application of this formula can be seen in examples like calculating tan(60º). We know that 60º is double 30º, and using the formula tan(2x) = (2tan(x)) / (1 - tan²(x)), with tan(30º) = 1/√3, we can find tan(60º) = √3.
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Formula: tan(2x) = (2tan(x)) / (1 - tan²(x))
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Derived from the ratio between sine and cosine: tan(x) = sin(x)/cos(x)
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Practical application: Calculation of tan(60º) using the double angle formula
Triple Angle Formula for Sine
The triple angle formula for sine is represented by sin(3x) = 3sin(x) - 4sin³(x). This formula is derived from trigonometric identities and allows us to express the sine of the triple of an angle in terms of the sine of the original angle.
To derive this formula, we can use the sum of angles identity and the double angle formula. We know that sin(3x) can be written as sin(2x + x). Using the angle sum formula, we obtain sin(3x) = sin(2x)cos(x) + cos(2x)sin(x). Substituting the double angle formulas for sin(2x) and cos(2x), we arrive at the formula sin(3x) = 3sin(x) - 4sin³(x).
The practical application of this formula can be observed in examples such as calculating sin(3 * 30º), where sin(90º) is known and can be verified as 1. This formula is especially useful in calculations involving multiples of 3 of known angles.
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Formula: sin(3x) = 3sin(x) - 4sin³(x)
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Derived from the angle sum identity: sin(2x + x)
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Practical application: Calculation of sin(90º) using the triple angle formula
To Remember
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Double Angle: Refers to the double of an angle and the associated formulas to calculate sine, cosine, and tangent.
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Triple Angle: Refers to the triple of an angle and the associated formulas to calculate sine, cosine, and tangent.
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Trigonometric Identity: Fundamental relationships between trigonometric functions used to derive formulas.
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Trigonometric Function: Functions that relate the angles of a triangle to its sides (sine, cosine, tangent).
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Sine: Trigonometric function that relates the length of the side opposite the angle with the hypotenuse of the triangle.
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Cosine: Trigonometric function that relates the length of the adjacent side to the angle with the hypotenuse of the triangle.
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Tangent: Trigonometric function that is the ratio between the sine and cosine of an angle.
Conclusion
During the lesson, we explored the double and triple angle formulas for sine, cosine, and tangent, which are essential tools of trigonometry. We understood how these formulas are derived from basic trigonometric identities and how they can be applied to simplify and solve complex problems involving double and triple angles.
The practical application of these formulas was demonstrated through numerical examples, such as calculating sin(60º) or cos(22.5º). These examples showed the importance of knowing and correctly applying the double and triple angle formulas to obtain accurate results in trigonometric calculations.
Understanding these formulas not only enriches students' mathematical knowledge but also opens doors to various practical applications in areas such as engineering, physics, and computer graphics. We encourage everyone to continue exploring the topic to develop advanced mathematical skills and apply them in academic and professional contexts.
Study Tips
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Review the practical examples discussed in class and try to solve additional problems using the double and triple angle formulas.
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Practice deriving the formulas from basic trigonometric identities to solidify your understanding of the relationships among trigonometric functions.
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Utilize additional resources such as textbooks, educational videos, and math apps to deepen your knowledge about the application of double and triple angle formulas in different contexts.
