In Exercises 1–8, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ.(3, 7)

Verified step by step guidance

1. Identify the coordinates of the point on the terminal side of angle \( \theta \) as \((x, y) = (3, 7)\).
2. Calculate the distance from the origin to the point, which is the hypotenuse \( r \) of the right triangle formed. Use the formula \( r = \sqrt{x^2 + y^2} \).
3. Determine the sine of \( \theta \) using \( \sin \theta = \frac{y}{r} \).
4. Determine the cosine of \( \theta \) using \( \cos \theta = \frac{x}{r} \).
5. Determine the tangent of \( \theta \) using \( \tan \theta = \frac{y}{x} \).

Key Concepts

Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are fundamental in trigonometry. They relate the angles of a triangle to the ratios of its sides. For a point (x, y) on the terminal side of an angle θ in standard position, these functions can be defined using the coordinates of the point and the radius (r) of the circle, where r = √(x² + y²).

Coordinates and Radius

In the context of trigonometry, a point (x, y) represents the coordinates of a point on the terminal side of an angle θ. The radius r is the distance from the origin to this point, calculated using the Pythagorean theorem: r = √(x² + y²). This radius is crucial for determining the values of the trigonometric functions, as they are defined in terms of x, y, and r.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric interpretation of the trigonometric functions, where the coordinates of any point on the circle correspond to the cosine and sine of the angle formed with the positive x-axis. Understanding the unit circle helps in visualizing and calculating the values of trigonometric functions for any angle.