Question

State the three Pythagorean identities.

Short answer

Answer: The three Pythagorean identities involving trigonometric functions are as follows: 1. sin^2(θ) + cos^2(θ) = 1 2. tan^2(θ) + 1 = sec^2(θ) 3. 1 + cot^2(θ) = csc^2(θ)

Step-by-step solution

Pythagorean Identity 1

The first Pythagorean identity relates the sine and cosine functions. In a unit circle, consider a point P with coordinates (x, y) and the angle θ in standard position. By definition of the trigonometric functions, cos(θ) = x and sin(θ) = y. According to the Pythagorean theorem, we have: x^2 + y^2 = 1 (since the radius of the unit circle is 1) Re-writing this expression in terms of trigonometric functions, we get: sin^2(θ) + cos^2(θ) = 1

Pythagorean Identity 2

The second Pythagorean identity relates the secant and tangent functions. To derive this identity, let's divide both sides of the first identity by cos^2(θ). We can be sure that cos^2(θ) ≠ 0, because θ here must not be equal to π/2 or 3π/2. The resulting equation will look like this: (sin^2(θ) + cos^2(θ)) / cos^2(θ) = 1 / cos^2(θ) Since sin(θ)/cos(θ) = tan(θ) and 1/cos(θ) = sec(θ), the above expression can be re-written as: tan^2(θ) + 1 = sec^2(θ)

Pythagorean Identity 3

The third Pythagorean identity relates the cosecant and cotangent functions. Similarly, we can derive this identity by dividing both sides of the first identity by sin^2(θ). We can be sure that sin^2(θ) ≠ 0, because θ here must not be equal to 0 or π. The resulting equation will look like this: (sin^2(θ) + cos^2(θ)) / sin^2(θ) = 1 / sin^2(θ) Since cos(θ)/sin(θ) = cot(θ) and 1/sin(θ) = csc(θ), the above expression can be re-written as: 1 + cot^2(θ) = csc^2(θ) So, the three Pythagorean identities are: 1. sin^2(θ) + cos^2(θ) = 1 2. tan^2(θ) + 1 = sec^2(θ) 3. 1 + cot^2(θ) = csc^2(θ)

Key concepts

Trigonometric Functions

When exploring the concept of trigonometric functions, it's imperative to understand their significance in mathematics. These functions, which include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot), provide relationships between the angles and sides of a triangle, and they extend these relationships to circular motion through the unit circle.

Each trigonometric function can be defined with respect to a specific angle, often denoted as \( \theta \). In the context of a right-angled triangle, for example, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Meanwhile, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

These functions go beyond the triangles and enter into an endless periodic behavior as seen when they're graphed. This periodic nature makes them incredibly useful in various fields like physics, engineering, and even music theory because repetitive patterns and cycles are fundamental to these domains.

Unit Circle

The unit circle is a fundamental concept that ties the trigonometric functions to circular motion. It is a circle with a radius of one unit, centered at the origin (0,0) of a coordinate plane. The unit circle allows us to define all the trigonometric functions precisely and to simply all triangle-based definitions to a circle-based framework.

As an angle \( \theta \) sweeps from the positive x-axis, the coordinates of a point on the circumference of the circle represent the cosine and sine of that angle. Specifically, if point P represents the angle \( \theta \) on the unit circle, and P has coordinates (x, y), then by definition \( \cos(\theta) = x \) and \( \sin(\theta) = y \).

One of the key reasons why the unit circle is so important is that it allows us to understand the trigonometric functions for all angle measurements, not just those that are easily represented by a right triangle. For angles greater than 90 degrees or less than 0 degrees, the unit circle provides a clear geometric interpretation.

Pythagorean Theorem

The Pythagorean theorem is a principle that sets the foundation for many concepts in mathematics, including the Pythagorean identities in trigonometry. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as \( a^2 + b^2 = c^2 \), where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the other sides.

When applied to a unit circle, the Pythagorean theorem becomes \( x^2 + y^2 = 1 \), where 'x' and 'y' are the coordinates of a point on the circle. This relationship is crucial as it generates the first Pythagorean identity when re-expressed using trigonometric functions. Specifically, since \( \cos(\theta) = x \) and \( \sin(\theta) = y \), substituting these into the circle equation yields \( \sin^2(\theta) + \cos^2(\theta) = 1 \), demonstrating one of the fundamental connections between right triangles and trigonometric functions.