Understanding the Trigonometric Function: Finding sinθ Using the Coordinates of a Point

In this article, we will delve into the process of finding the value of sinθ when a point is given on the terminal arm of an angle. Let's explore the steps and the mathematical reasoning behind it.

Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles. While it is often used in geometry and physics, trigonometric functions can also be defined using the coordinates of points in a plane. This is particularly useful when working with polar coordinates and angle measurements.

Problem: The Point (-4, 3) on the Terminal Arm of Angle θ

Consider a point (-4, 3) located on the terminal arm of an angle θ. Our goal is to calculate the value of sinθ.

Step-by-Step Solution

To find sinθ, we need to follow the fundamental trigonometric relationships. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

Calculating the Hypotenuse

First, we need to determine the length of the hypotenuse, r. This is the distance from the origin to the point (-4, 3).

Using the distance formula:

r2 -42 32

Simplifying:

r2 16 9 25 ? r 25 5

Calculating sinθ

Now that we have the hypotenuse, we can calculate the sine of angle θ using the formula:

sinθ opposite hypotenuse 3 5

Therefore, the value of sinθ is:

sinθ 0.6

Conclusion

This example demonstrates how to use basic trigonometric principles and the distance formula to find the sine of an angle given a point on its terminal arm. Understanding these concepts is crucial for solving more complex trigonometric problems and has numerous applications in fields such as engineering, physics, and computer science.

Other Trigonometric Ratios

The sine is just one of the trigonometric ratios. Other important ratios include cosine and tangent. Knowing how to calculate each of these can help you solve a wide array of trigonometric problems.

Practice Problems

Try finding cosθ and tanθ for the same point (-4, 3).

Solution for cosθ

cosθ adjacent hypotenuse -4 5

cosθ -0.8

Solution for tanθ

tanθ opposite adjacent 3 -4

tanθ -0.75

Key Takeaways

Trigonometric functions can be defined using the coordinates of points. The distance formula is used to find the hypotenuse. The sine of an angle is the ratio of the opposite side to the hypotenuse. Practice problems can help reinforce understanding of trigonometric ratios.