Problem 256 - Precalculus

Question

When a right triangle with a hypotenuse of 1 is placed in the unit circle, which sides of the triangle correspond to the \(x-\) and \(y\) -coordinates?

Step-by-Step Solution

Answer

The x-coordinate is \(\cos(\theta)\), and the y-coordinate is \(\sin(\theta)\).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. Any triangle inscribed in this circle would have a hypotenuse equal to the circle's radius, which is 1.

Step 2: Identify the Triangle's Placement

Placing the right triangle in the unit circle typically involves having one vertex at the origin, one vertex on the x-axis, and the third vertex on the circle itself. This setup allows the triangle to align with the coordinate axes.

Step 3: Apply Pythagorean Theorem

In the right triangle, use the Pythagorean theorem. The theorem states that for a right triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), the relationship is \(a^2 + b^2 = c^2\). Here, \(c = 1\).

Step 4: Relate Triangle Sides to Coordinates

The two legs of the triangle correspond to the x and y coordinates. If the angle at the origin is \(\theta\), then the leg along the x-axis has a length \(\cos(\theta)\), and the leg along the y-axis has a length \(\sin(\theta)\).

Step 5: Conclusion

Therefore, the sides of the right triangle correspond to the x-coordinate being \(\cos(\theta)\) and the y-coordinate being \(\sin(\theta)\) when placed in the unit circle.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring 90 degrees. This angle is known as a right angle. The side opposite this right angle is the hypotenuse, which is always the longest side of the triangle. The other two sides of the triangle are known as legs or catheti.
Understanding a right triangle is essential when working with the unit circle because the triangle's properties can help determine the coordinates on the unit circle.
Here's why right triangles are important in this scenario:

• They allow us to use trigonometric functions, like sine and cosine.
• The placement of the triangle inside the unit circle helps us relate its sides to the circle's radius.
• They help to visualize and solve problems in the coordinate plane using the Pythagorean theorem.

Thus, right triangles form a fundamental basis for understanding the geometric relationships in the unit circle.

Pythagorean Theorem

The Pythagorean theorem is a principle that directly applies to right triangles. It states: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. In the context of the unit circle, the hypotenuse equals 1 because it is the radius of the unit circle.
This means: \[ a^2 + b^2 = 1^2 \] This simple yet powerful equation allows us to find the unknown side lengths of the triangle.
Applied to the unit circle:

• It means that the sum of the squares of the x-coordinate and y-coordinate must equal 1.
• This helps in finding the exact position of a point on the circle.

The Pythagorean theorem thus serves as a crucial link between algebraic and geometric perspectives.

Trigonometric Functions

Trigonometric functions are essential mathematical tools that relate angles to side lengths in right triangles. When dealing with the unit circle, the primary trigonometric functions used are sine and cosine. These functions are defined based on the angle \(\theta\) formed at the origin of the coordinate plane.
In the context of the unit circle:

• The cosine of an angle \(\theta\) gives the x-coordinate of the point on the unit circle: \(\cos(\theta)\).
• The sine of an angle \(\theta\) gives the y-coordinate: \(\sin(\theta)\).

This means every point on the unit circle can be described using these functions. Trigonometric functions thus enable us to easily determine the location of points using angles rather than just Cartesian coordinates. They also help in solving complex problems involving periodic functions or wave-like scenarios, which often arise in physics and engineering.

Coordinate Plane

The coordinate plane is a fundamental concept in geometry and algebra. It's a two-dimensional plane formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis), which are perpendicular to each other. The point of intersection of these axes is called the origin, represented by the point (0,0).
Understanding the coordinate plane is vital in visualizing and solving problems involving the unit circle:

• It provides a framework for placing geometric figures, such as circles and triangles, in a logical and consistent manner.
• Coordinates on this plane, such as \((x, y)\), give us precise location details in terms of distance from the origin on both axes.
• Any point's coordinates can be linked directly to trigonometric functions when working with the unit circle.

Therefore, the coordinate plane allows us to explore relationships between algebraic expressions and geometric figures clearly and effectively.