Trigonometric Functions and the Pythagorean Theorem

Trigonometric Functions

Introduction to Trigonometric Functions

Trigonometric functions are fundamental mathematical tools used to relate the angles and sides of triangles, especially right triangles. These functions are based on the relationships within right triangles and are essential for understanding geometry, physics, and engineering applications.

• Trigonometric functions describe the ratios between the sides of a right triangle relative to its angles.

• They are defined using the Pythagorean Theorem, which relates the lengths of the sides of a right triangle.

• These functions are always based on the right triangle, where one angle is 90 degrees.

The Pythagorean Theorem

The Pythagorean Theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the sum of the squares of the two legs is equal to the square of the hypotenuse.

• Formula: $a^2 + b^2 = c^2$

• a and b are the lengths of the legs.

• c is the length of the hypotenuse (the side opposite the right angle).

• This theorem is used to derive the formulas for calculating trigonometric functions.

Example: For a triangle with sides 3, 4, and 5, $3^2 + 4^2 = 5^2$ or $9 + 16 = 25$.

Common Pythagorean Triples
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
9, 40, 41

Trigonometric Functions in the Coordinate Plane

Trigonometric functions can be defined using a point P(x, y) in the coordinate plane. By dropping a perpendicular from P to the x-axis, a right triangle is formed with the origin.

• The hypotenuse is $r = \sqrt{x^2 + y^2}$.

• The sides are the x and y coordinates.

• Trigonometric functions are defined as ratios involving x, y, and r.

Example: If P is at (3, 4), then $r = \sqrt{3^2 + 4^2} = 5$.

Definitions of Trigonometric Functions

Each of the six trigonometric functions is defined as a ratio of the sides of a right triangle:

• Sine: $\sin \theta = \frac{y}{r}$

• Cosine: $\cos \theta = \frac{x}{r}$

• Tangent: $\tan \theta = \frac{y}{x}$

• Cosecant: $\csc \theta = \frac{r}{y}$

• Secant: $\sec \theta = \frac{r}{x}$

• Cotangent: $\cot \theta = \frac{x}{y}$

These ratios allow us to calculate angles and side lengths in right triangles and are foundational for further study in trigonometry.

Additional info: The notes provide a basic introduction to trigonometric functions, the Pythagorean Theorem, and their application in the coordinate plane. The table of Pythagorean triples is included for reference, and the image illustrates the geometric setup for defining trigonometric functions.