Summary Tradisional | Trigonometric Lines of 75º, 15º, 22.5º

Contextualization

Trigonometry is one of the oldest and most essential branches of mathematics, with real-world applications across various fields like engineering, physics, and architecture. Specific angles like 75º, 15º, and 22.5º frequently appear in precise calculations, whether in designing complex architectural structures or in areas like navigation and astronomy. A solid grasp of the trigonometric functions associated with these angles sharpens problem-solving skills and fosters better mathematical reasoning.

These angles have unique trigonometric properties that can be derived using sum and difference angle formulas, along with the half-angle formula. Such formulas are critical for calculating the sine and cosine values of these angles. For instance, astronomers depend on these angles to pinpoint the locations of stars and planets, while engineers apply them in structural design. Hence, mastering these trigonometric properties is vital for various practical applications in everyday life.

To Remember!

Trigonometric Functions of 75º

To find the trigonometric functions of 75º, we employ the angle sum formula. Specifically, 75º can be expressed as the sum of 45º and 30º. The relevant formulas are:

Sine: (\sin(75º) = \sin(45º + 30º) = \sin 45º \cos 30º + \cos 45º \sin 30º) Cosine: (\cos(75º) = \cos(45º + 30º) = \cos 45º \cos 30º - \sin 45º \sin 30º)

Substituting the known values of the trigonometric functions for 45º and 30º yields:

(\sin 75º = \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) + \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}) (\cos 75º = \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4})

Breaking down 75º into 45º + 30º simplifies calculations and allows direct application of the angle sum formulas, making it easier to derive the exact trigonometric functions.

• 75º can be expressed as 45º + 30º.

• The angle sum formula is used to derive sine and cosine.

• Final values are obtained by substituting the known trigonometric functions.

Trigonometric Functions of 15º

To find the trigonometric functions of 15º, we utilize the angle difference formula. Specifically, 15º can be described as the difference between 45º and 30º. The relevant formulas are:

Sine: (\sin(15º) = \sin(45º - 30º) = \sin 45º \cos 30º - \cos 45º \sin 30º) Cosine: (\cos(15º) = \cos(45º - 30º) = \cos 45º \cos 30º + \sin 45º \sin 30º)

By substituting the known values of the trigonometric functions for 45º and 30º, we obtain:

(\sin 15º = \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}) (\cos 15º = \left( \frac{\sqrt{2}}{2} \right) \left( \frac{\sqrt{3}}{2} \right) + \left( \frac{\sqrt{2}}{2} \right) \left( \frac{1}{2} \right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4})

Breaking down 15º into 45º - 30º allows us to directly apply the angle difference formulas, simplifying the process to derive the exact trigonometric functions.

• 15º can be expressed as 45º - 30º.

• The angle difference formula is used to derive sine and cosine.

• Final values are obtained by substituting the known trigonometric functions.

Trigonometric Functions of 22.5º

To calculate the trigonometric functions of 22.5º, we apply the half-angle formula. In this case, 22.5º is half of 45º. The formulas involved are:

Sine: (\sin(22.5º) = \sin \left( \frac{45º}{2} \right) = \sqrt{\frac{1 - \cos 45º}{2}}) Cosine: (\cos(22.5º) = \cos \left( \frac{45º}{2} \right) = \sqrt{\frac{1 + \cos 45º}{2}})

As 22.5º is located in the first quadrant, we opt for the positive sign. Substituting the known values of the trigonometric functions for 45º gives us:

(\sin 22.5º = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}) (\cos 22.5º = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \frac{\sqrt{2 + \sqrt{2}}}{2})

Using the half-angle formula allows for a simplified process in calculating the exact sine and cosine values of 22.5º.

• 22.5º is half of 45º.

• The half-angle formula is used to derive sine and cosine.

• Final values are obtained by substituting the known trigonometric functions.

Practical Applications of Trigonometric Functions

The trigonometric functions of 75º, 15º, and 22.5º find numerous practical applications across fields such as engineering, physics, and architecture. In civil engineering, for example, these angles are often used in the design and construction of structures, like bridges and buildings, where accuracy in calculations is crucial for safety and stability.

In physics, these trigonometric functions come into play in calculations regarding waves and oscillations, such as analyzing vibrations and resonances. This knowledge is essential for developing equipment and systems that require precise and controlled movements.

Architecture utilizes these angles in creating both aesthetic and functional designs, such as domes and arches, where mathematical accuracy is key to achieving visual balance and structural robustness. Furthermore, in computer graphics, angles like 75º, 15º, and 22.5º are used to build precise and realistic three-dimensional models.

Understanding the trigonometric functions relevant to these angles enables more effective solutions to practical challenges, marrying mathematical theory with real-world applications and leading to innovative, safe solutions.

• Applied in the design and construction of structures in civil engineering.

• Used for calculations involving waves and oscillations in physics.

• Important for aesthetic and functional design in architecture.

• Utilized in computer graphics for creating accurate three-dimensional models.

Key Terms

• Sine: A trigonometric function that expresses the ratio of the opposite side to the hypotenuse of a right-angled triangle.

• Cosine: A trigonometric function that conveys the ratio of the adjacent side to the hypotenuse of a right-angled triangle.

• Angle Sum: Trigonometric formulas that facilitate calculating functions of an angle represented as the sum of two angles.

• Angle Difference: Trigonometric formulas that enable calculations of functions for an angle described as the difference between two angles.

• Half-Angle: Trigonometric formulas that facilitate the calculation of functions for an angle that equals half another angle.

• Trigonometry: A mathematical discipline exploring the relationships between the angles and sides of triangles.

• Trigonometric Functions: The computed values of trigonometric functions (sine, cosine) for specific angles.

Important Conclusions

In this lesson, we delved into the trigonometric functions of the angles 75º, 15º, and 22.5º by applying the formulas for angle sum, angle difference, and half-angle correspondingly. We discovered how to accurately compute the sine and cosine values of these angles, which is vital for tackling intricate mathematical problems in real-world scenarios.

Additionally, we underscored the practical applications of these trigonometric functions in various domains including engineering, physics, architecture, and computer graphics. Grasping these applications aids in illustrating the material's significance, inspiring a deeper investigation into the subject matter.

Ultimately, we reaffirm the necessity of mastering these mathematical tools to cultivate strong logical-mathematical reasoning and their application in genuine contexts, whether in overcoming academic challenges or in professional environments.

Study Tips

• Review the formulas for angle sum, angle difference, and half-angle, practicing their application with different examples.

• Make use of supplementary resources such as videos and online exercises to visualize and deepen your understanding of practical applications of trigonometric functions.

• Form study groups to engage in discussions and resolve practical problems involving the trigonometric functions of 75º, 15º, and 22.5º, sharing knowledge and strategies.