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.(-4, 3)

Verified step by step guidance

1. Identify the coordinates of the point on the terminal side of angle \( \theta \), which are \((-4, 3)\).
2. Calculate the radius \( r \) using the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \), where \( x = -4 \) and \( y = 3 \).
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 \( \tan \theta = \frac{y}{x} \), and find the reciprocal functions: \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).

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).

Coordinates and the Unit Circle

In trigonometry, the coordinates of a point on the terminal side of an angle θ can be visualized using the unit circle, where the radius is 1. For any point (x, y), the angle θ can be determined, and the coordinates help in finding the values of the trigonometric functions. The unit circle provides a framework for understanding how these functions behave as angles change.

Pythagorean Theorem

The Pythagorean theorem is essential for calculating 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 is crucial for determining the exact values of the trigonometric functions based on the coordinates of the point.