Pythagorean Identities

Introduction

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Understanding these relationships helps us in many fields such as physics, engineering, and architecture. One of the most important concepts in trigonometry is the Pythagorean identities. These identities are derived from the Pythagorean theorem and play an important role in simplifying complex trigonometric expressions.

Pythagorean theorem

Before we dive into the Pythagorean identity, let's briefly revisit the Pythagorean Theorem. This is a fundamental relationship between the three sides of a right triangle in Euclidean geometry. The theorem states that:

a² + b² = c²

Here, c is the hypotenuse, the side opposite the right angle, and a and b are the other two sides of the triangle.

Basic trigonometric ratios

In trigonometry, the most common functions we deal with are sine (sin), cosine (cos), and tangent (tan). These functions are used to relate the angles and sides of a triangle. They are defined as follows for any angle θ in a right triangle:

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent

Defining the Pythagorean identity

The Pythagorean identities are derived from the three main trigonometric functions and are related to the Pythagorean Theorem. These identities express fundamental interrelationships between the squares of the sine, cosine, and tangent functions.

Elementary Pythagorean identity

This identity can be expressed as follows:

sin²(θ) + cos²(θ) = 1

This is directly derived from the Pythagorean theorem. To understand this:

Here, a, b and c correspond to sine, cosine, and 1 respectively when considering a unit circle where the hypotenuse is always 1.

Secondary Pythagorean identity

In addition to the elementary identities, there are two other important identities derived using the tangent function and the reciprocal identity:

The first one is this:

1 + tan²(θ) = sec²(θ)

This can be obtained by dividing the primary identity by cos²(θ).

The second is:

1 + cot²(θ) = csc²(θ)

This can be obtained by dividing the primary identity by sin²(θ).

Visualizing with the unit circle

To understand these identities further, it is useful to look at them on the unit circle. The unit circle is a circle with radius one, centered at the origin of the coordinate plane. When we construct an angle in the unit circle, the x-coordinate of the point on the circle is equal to cos(θ), and the y-coordinate is equal to sin(θ).

The hypotenuse of a line from the origin to any point on the circle will be 1, forming a right triangle with the x-axis and perpendicular to the x-axis from the point on the circle. The Pythagorean theorem confirms that in these cases, sin²(θ) + cos²(θ) = 1.

Examples and applications of Pythagorean identities

Let's look at some examples that show how the Pythagorean identities are used to solve trigonometric problems:

Example 1

Suppose you know sin(θ) = 3/5, and you need to find cos(θ) Since you know:

sin²(θ) + cos²(θ) = 1

Substitute the value of sin(θ) that we know:

(3/5)² + cos²(θ) = 1

Simplifying it further:

9/25 + cos²(θ) = 1

Solution of cos²(θ):

cos²(θ) = 1 - 9/25
cos²(θ) = 16/25
cos(θ) = ±√(16/25)
cos(θ) = ±4/5

Cosine can be positive or negative depending on the quadrant where the angle θ is located.

Example 2

Given tan(θ) = 2, find sec(θ).

By identity:

1 + tan²(θ) = sec²(θ)

Plug in the given values:

1 + 2² = sec²(θ)
1 + 4 = sec²(θ)
sec²(θ) = 5
sec(θ) = ±√5

Again, remember that the secant can be positive or negative depending on the quadrant.

Proof of the Pythagorean identity

Let's set up the proofs for these identities so we understand how we derive them.

Proof of the elementary Pythagorean identity

Consider a right triangle where:

• Hypotenuse c = 1 (unit circle)
• The opposite side is sin(θ)
• The adjacent side is cos(θ)

Apply the Pythagorean theorem:

sin²(θ) + cos²(θ) = 1²

This gives me recognition.

Proof of the secondary Pythagorean identity

For 1 + tan²(θ) = sec²(θ):

Starting with:

sin²(θ) + cos²(θ) = 1

Divide each term by cos²(θ):

(sin²(θ) / cos²(θ)) + 1 = 1 / cos²(θ)
tan²(θ) + 1 = sec²(θ)

For 1 + cot²(θ) = csc²(θ):

Restart:

sin²(θ) + cos²(θ) = 1

Divide each term by sin²(θ):

1 + (cos²(θ) / sin²(θ)) = 1 / sin²(θ)
1 + cot²(θ) = csc²(θ)

Why are Pythagorean identities important?

Pythagorean identities allow us to transform complex trigonometric expressions into simpler ones. They are essential tools for solving trigonometric equations, proving other identities, and understanding the properties of trigonometric functions.

In practical applications, these identities ensure that calculations involving angles remain consistent and accurate, which is especially useful in fields such as navigation, physics, and engineering, which require precise measurements.

Conclusion

Understanding the Pythagorean identities is important for anyone studying trigonometry. These identities provide a basis for solving problems, simplifying expressions, and understanding the circular nature of trigonometric functions. By mastering these identities, you will gain a more in-depth knowledge of how angles and triangles relate to each other, which will increase your mathematical ability and problem-solving abilities.

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