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.(5, -5)

Verified step by step guidance

1. insert step 1: Identify the coordinates of the point on the terminal side of angle \( \theta \), which are \((x, y) = (5, -5)\).
2. insert step 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. insert step 3: Determine the six trigonometric functions using the definitions: \( \sin(\theta) = \frac{y}{r} \), \( \cos(\theta) = \frac{x}{r} \), \( \tan(\theta) = \frac{y}{x} \), \( \csc(\theta) = \frac{r}{y} \), \( \sec(\theta) = \frac{r}{x} \), and \( \cot(\theta) = \frac{x}{y} \).
4. insert step 4: Substitute the values of \( x \), \( y \), and \( r \) into the trigonometric function definitions to find their exact values.
5. insert step 5: Simplify each trigonometric function to its simplest form, ensuring to rationalize denominators if necessary.

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) calculated as r = √(x² + y²).

Coordinate System and Angle Measurement

In trigonometry, angles are often measured in standard position, where the vertex is at the origin and the initial side lies along the positive x-axis. The terminal side of the angle is determined by the coordinates of a point (x, y). Understanding the quadrant in which the angle lies is crucial, as it affects the signs of the trigonometric functions derived from the coordinates.

Pythagorean Theorem

The Pythagorean theorem is essential for finding the radius (r) when given a point (x, y). It states that in a right triangle, the square of the hypotenuse (r) is equal to the sum of the squares of the other two sides (x and y). This relationship allows us to compute r as r = √(x² + y²), which is necessary for calculating the trigonometric functions based on the coordinates of the point.