Problem 105 - Algebra and Trigonometry

Question

If you are given a point on the terminal side of angle \(\theta .\) explain how to find \(\sin \theta\)

Step-by-Step Solution

Answer

Given a point (x, y) on the terminal side of angle \(\theta\), \(\sin \theta\) can be found by dividing the y-coordinate of the point by the length of the hypotenuse, giving \(\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}\).

Step 1: Determine the position of the point

Locate the given point in the complex plane, forming an angle \(\theta\) with the positive direction of the x-axis.

Step 2: Draw a right triangle

From the origin, draw a line segment perpendicular to the x-axis such that it intersects with the line passing through origin and the given point. This forms a right triangle.

Step 3: Identify the sides of the triangle

The horizontal leg of the triangle is along the x-axis, and the vertical leg is the line we just drew perpendicular to the x-axis. The hypotenuse is the line from the origin to the given point.

Step 4: Find the length of the opposite side

The length of the side opposite to the angle \(\theta\) can be found using the coordinates of the given point. If the point is (x, y), then the length of the opposite side is y.

Step 5: Calculate the length of the hypotenuse

The length of the hypotenuse can be found using the Pythagorean theorem. If the coordinates of the point are (x, y), then the length of the hypotenuse is \(\sqrt{x^2 + y^2}\).

Step 6: Calculate the value of \(\sin \theta\)

The value of \(\sin \theta\) is the length of the side opposite the angle divided by the length of the hypotenuse. Hence, \(\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}\).

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry is a method of solving problems involving triangles with one right angle, which is essential in many aspects of math and science. In a right triangle, there are three sides: the hypotenuse, the opposite side, and the adjacent side. Each of these sides has a specific relation to the other, which allows us to determine various trigonometric functions, such as sine, cosine, and tangent.
By understanding the properties of these functions, we can solve for unknown angles or sides.

• Hypotenuse: The longest side of the triangle, opposite the right angle.
• Opposite Side: The side opposite the angle you are examining (not the right angle).
• Adjacent Side: The side next to the angle you are examining (not the hypotenuse).

In practice, right triangle trigonometry is used to find angles or side lengths when given certain values, such as in navigation, architecture, and various fields of engineering.

Sine Function

The sine function is a fundamental component of trigonometry, particularly when working with right triangles. It is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Hence, for any angle \( \theta \) in a right triangle, we define \( \sin \theta \) as \( \sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \).
The sine function helps us connect angles to side lengths, making it immensely useful in various applications such as physics, engineering, and even computer graphics.
Moreover, the sine function does not only apply to right triangles but can also be extended to any angle using the unit circle. This allows sine to be used to describe wave patterns, such as sound waves, and to solve various periodic functions in science and technology.

Coordinate System in Trigonometry

A coordinate system in trigonometry allows us to precisely describe a point's position in a plane, which is pivotal in solving trigonometric problems. Particularly, the Cartesian coordinate system uses two axes – the horizontal (x-axis) and the vertical (y-axis) – to define a point's position with a pair of numbers, usually written as \((x, y)\).
When dealing with trigonometric functions, this system helps in visualizing angles and triangles. For example, when finding \( \sin \theta \) from a point \((x, y)\) in a coordinate plane, this point is typically viewed from the origin \((0, 0)\) acting as the vertex of an angle formed with the x-axis.
In the context of trigonometry, the coordinate system simplifies understanding by providing a visual representation of sine, cosine, and tangent values. This is because each trigonometric function can be associated with a certain segment in a right triangle formed by the coordinate point, thus bridging algebra and geometry for comprehensive problem-solving.