Question

Explain how a point \(P(x, y)\) on a circle of radius \(r\) determines an angle \(\theta\) and the values of the six trigonometric functions at \(\theta\)

Short answer

Question: A point P(x, y) is on a circle of radius r. Determine the values of the six trigonometric functions at the angle θ formed by the line connecting the origin and the point P(x, y) with the x-axis. Answer: Using the definitions of the six trigonometric functions, we can find: 1. \(\sin(\theta) = \frac{y}{r}\) 2. \(\cos(\theta) = \frac{x}{r}\) 3. \(\tan(\theta) = \frac{y}{x}\) 4. \(\csc(\theta) = \frac{r}{y}\) 5. \(\sec(\theta) = \frac{r}{x}\) 6. \(\cot(\theta) = \frac{x}{y}\)

Step-by-step solution

Draw the right triangle

Draw a right triangle connecting the origin, the point P(x, y), and the projection of P onto the x-axis. Label the angle at the origin as θ and the sides of the triangle as x, y, and r.

Define the trigonometric functions

Recall the definitions of the six trigonometric functions based on the sides of a right triangle: 1. \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\) 2. \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\) 3. \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\) 4. \(\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}\) 5. \(\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}}\) 6. \(\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}\)

Calculate the values of the trigonometric functions

Apply the definitions of the trigonometric functions to our triangle with sides x, y, and r. Here, x and y are the "adjacent" and "opposite" sides respectively, while r is the "hypotenuse". Calculate the values as follows: 1. \(\sin(\theta) = \frac{y}{r}\) 2. \(\cos(\theta) = \frac{x}{r}\) 3. \(\tan(\theta) = \frac{y}{x}\) 4. \(\csc(\theta) = \frac{r}{y}\) 5. \(\sec(\theta) = \frac{r}{x}\) 6. \(\cot(\theta) = \frac{x}{y}\) Now we have determined the angle θ and the values of the six trigonometric functions at θ with respect to the point P(x, y) on the circle of radius r.

Key concepts

Circle

A circle is a collection of points that are all at the same distance from a central point. The fixed distance from the center to any point on the circle is called the radius.

When working with trigonometric functions, imagine a circle is drawn on a coordinate plane with its center at the origin (0, 0). This is known as the unit circle. It serves as a handy tool for understanding these functions.

In a standard unit circle, every point on the circumference can be described by its coordinates

• The x-coordinate represents the cosine of the angle formed with the positive x-axis.
• The y-coordinate represents the sine of the angle.

Thus, points on a circle are closely connected to trigonometric functions, linking geometry and trigonometry together.

Right Triangle

A right triangle contains a 90-degree angle, making it a perfect shape for exploring trigonometric functions. By connecting the origin, a point on the circle, and its projection on the x-axis, we form this kind of triangle.

The sides of the right triangle are described as:

• The side opposite the angle of interest (the angle at the origin), which is vertical.
• The side adjacent to the angle of interest, which is horizontal.
• The hypotenuse, which is the diagonal side connecting the point on the circle to the origin, with a length equal to the radius of the circle.

Using right triangles helps define the trigonometric functions by relating their side lengths to specific angles.

Sine

The sine function is a fundamental trigonometric function that relates an angle in a right triangle to the ratio of the lengths of two sides. Specifically, sine of an angle θ is defined as the ratio of the opposite side to the hypotenuse. In mathematical terms:

\(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}.\)

When a point lies on a circle with radius r, and forms an angle θ with the x-axis, sin(θ) becomes y/r, where y is the vertical coordinate of the point. This definition directly connects the geometric space of the circle to the algebraic world of functions, making sine a bridge between these two perspectives.

Cosine

The cosine function, like sine, is a fundamental part of trigonometry. It describes how the adjacent side of a right triangle relates to the hypotenuse. Cosine of an angle θ is defined as the ratio of the adjacent side to the hypotenuse:

\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

In the context of a point on a circle of radius r, forming an angle θ with the x-axis, this becomes \(\cos(\theta) = \frac{x}{r},\)where x is the horizontal distance from the origin to the point.

Cosine is crucial as it translates circular motion and angles into linear, understandable terms. This encapsulates how trigonometric functions tie diverse concepts in mathematics together.