Understanding the Sine Function and Its Periodicity: Exploring sin 540° and sin 5085°
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. The sine function, in particular, is a fundamental component of trigonometry, and understanding its periodic nature is crucial for solving a wide range of mathematical problems. This article aims to explore the exact values of sin 540° and sin 5085°, and how the periodicity of the sine function simplifies these calculations.
Understanding the Sine Function
The sine function is a periodic function with a period of 360°. This means that for any angle theta, the sine of the angle is the same as the sine of that angle plus any integer multiple of 360°. Mathematically, this can be represented as:
sin(theta 360^circ k) sin(theta)
where k is any integer. This periodicity allows us to simplify the calculation of the sine of large angles by reducing them to an equivalent angle within the first cycle from 0° to 360°.
The Exact Value of sin 540°
To find the exact value of sin 540°, we first simplify the angle using the periodicity of the sine function.
sin 540° sin(540° - 360°) sin 180°
Since sin 180° 0, it follows that:
sin 540° 0
The Exact Value of sin 5085°
To find the exact value of sin 5085°, we can use the same periodicity approach. First, we divide 5085° by 360° to find how many full cycles there are:
frac{5085}{360} 14.125
This means sin 5085° is equivalent to sin 45°, because 5085° - 14(360°) 45°.
sin 5085° sin 45° frac{sqrt{2}}{2}
Deriving the Exact Values: A Step-by-Step Guide
Using the Periodic Nature of Sine
Another way to approach these calculations is by recognizing that a full cycle of the sine function is 360°. Therefore, we can use the periodicity rule to simplify the angles:
sin 540° 2k 360°,] where (k) is an integer.
This simplifies the angle to:
sin 540° sin(180° 360° times 1) sin 180° 0
The Double Angle Formula for Sine
The double angle formula for sine is a useful identity that can be applied to simplify the calculation of sin 540° and sin 5085°:
sin 2A 2sin Acos A
Applying this formula to 540° 2 times 270°, we get:
sin 540° sin(2 times 270°) 2sin 270°cos 270°
Since sin 270° -1 and cos 270° 0, we have:
2sin 270°cos 270° 2(-1)(0) 0
Thus, sin 540° 0.
Similarly, for sin 5085°, we can use the formula:
sin 5085° sin(2 times 2542.5°) 2sin 2542.5°cos 2542.5°
Since sin 5085° sin(14 times 360° 45°) sin 45° and sin 45° frac{sqrt{2}}{2}, it follows that:
sin 5085° frac{sqrt{2}}{2}
Conclusion and Further Applications
Understanding the periodicity of the sine function is essential for solving trigonometric problems. The exact values of sin 540° and sin 5085° are 0 and frac{sqrt{2}}{2}, respectively. These values can be applied to various mathematical and real-world scenarios, such as in engineering, physics, and navigation.
For a deeper understanding of trigonometric functions and their periodicity, further study is recommended. This foundation will aid in solving more complex trigonometric problems and equations.