Definition of Sine and Cosine

Oct 13, 2025 by miodeya.com 41 views

Introduction

In trigonometry, the Pythagorean identity is a fundamental concept that relates the sine and cosine functions. The Pythagorean identity states that for any angle $\theta$ , the sum of the squares of the sine and cosine of the angle is equal to 1. In this article, we will prove that $\sin^2 45^{\circ} + \cos^2 45^{\circ} = 1$ using the unit circle and the definition of sine and cosine.

The Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The unit circle is used to define the trigonometric functions sine and cosine. The sine of an angle $\theta$ is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle formed by the angle. The cosine of an angle $\theta$ is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle formed by the angle.

Definition of Sine and Cosine

The sine and cosine functions can be defined using the unit circle as follows:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

Proof of the Pythagorean Identity

To prove that $\sin^2 45^{\circ} + \cos^2 45^{\circ} = 1$ , we can use the definition of sine and cosine and the unit circle.

Let's consider a right triangle with an angle of $45^{\circ}$ . The triangle has two sides of equal length, which we can call $x$ . The hypotenuse of the triangle has a length of $\sqrt{2}x$ .

Using the definition of sine and cosine, we can write:

$\sin 45^{\circ} = \frac{x}{\sqrt{2}x} = \frac{1}{\sqrt{2}}$
$\cos 45^{\circ} = \frac{x}{\sqrt{2}x} = \frac{1}{\sqrt{2}}$

Now, we can square both sides of the equations to get:

$\sin^2 45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}$
$\cos^2 45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}$

Finally, we can add the two equations together to get:

$\sin^2 45^{\circ} + \cos^2 45^{\circ} = \frac{1}{2} + \frac{1}{2} = 1$

Conclusion

In this article, we proved that $\sin^2 45^{\circ} + \cos^2 45^{\circ} = 1$ using the unit circle and the definition of sine and cosine. The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. It is used to solve problems involving right triangles and is a key concept in calculus and other areas of mathematics.

Applications of the Pythagorean Identity

The Pythagorean identity has many applications in mathematics and other fields. Some of the applications include:

Trigonometry: The Pythagorean identity is used to solve problems involving right triangles, including finding the lengths of sides and the measures of angles.
Calculus: The Pythagorean identity is used to find the derivatives of trigonometric functions and to solve problems involving optimization and physics.
Engineering: The Pythagorean identity is used to solve problems involving right triangles and to find the lengths of sides and the measures of angles in engineering applications.
Physics: The Pythagorean identity is used to solve problems involving right triangles and to find the lengths of sides and the measures of angles in physics applications.

Examples of the Pythagorean Identity

Here are some examples of the Pythagorean identity:

$\sin^2 30^{\circ} + \cos^2 30^{\circ} = 1$
$\sin^2 60^{\circ} + \cos^2 60^{\circ} = 1$
$\sin^2 90^{\circ} + \cos^2 90^{\circ} = 1$

Solving Problems Using the Pythagorean Identity

Here are some examples of how to solve problems using the Pythagorean identity:

Example 1: Find the length of the hypotenuse of a right triangle with an angle of $45^{\circ}$ and a side length of 3.

Using the Pythagorean identity, we can write:

$\sin^2 45^{\circ} + \cos^2 45^{\circ} = 1$

We can substitute the values of $\sin 45^{\circ}$ and $\cos 45^{\circ}$ to get:

$\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 1$

Simplifying the equation, we get:

$\frac{1}{2} + \frac{1}{2} = 1$

Therefore, the length of the hypotenuse is $\sqrt{2} \times 3 = 3\sqrt{2}$ .
Example 2: Find the measure of an angle in a right triangle with a side length of 4 and a hypotenuse of 5.

Using the Pythagorean identity, we can write:

$\sin^2 \theta + \cos^2 \theta = 1$

We can substitute the values of $\sin \theta$ and $\cos \theta$ to get:

$\left(\frac{4}{5}\right)^2 + \left(\frac{3}{5}\right)^2 = 1$

Simplifying the equation, we get:

$\frac{16}{25} + \frac{9}{25} = 1$

Therefore, the measure of the angle is $\sin^{-1} \left(\frac{4}{5}\right)$ .

Conclusion

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. It states that for any angle $\theta$ , the sum of the squares of the sine and cosine of the angle is equal to 1.

Q: How is the Pythagorean identity used in trigonometry?

A: The Pythagorean identity is used to solve problems involving right triangles, including finding the lengths of sides and the measures of angles. It is also used to find the derivatives of trigonometric functions and to solve problems involving optimization and physics.

Q: What are some examples of the Pythagorean identity?

A: Some examples of the Pythagorean identity include:

$\sin^2 30^{\circ} + \cos^2 30^{\circ} = 1$
$\sin^2 60^{\circ} + \cos^2 60^{\circ} = 1$
$\sin^2 90^{\circ} + \cos^2 90^{\circ} = 1$

Q: How is the Pythagorean identity used in calculus?

A: The Pythagorean identity is used to find the derivatives of trigonometric functions and to solve problems involving optimization and physics. It is also used to solve problems involving right triangles and to find the lengths of sides and the measures of angles.

Q: What are some applications of the Pythagorean identity?

A: Some applications of the Pythagorean identity include:

Trigonometry: The Pythagorean identity is used to solve problems involving right triangles, including finding the lengths of sides and the measures of angles.
Calculus: The Pythagorean identity is used to find the derivatives of trigonometric functions and to solve problems involving optimization and physics.
Engineering: The Pythagorean identity is used to solve problems involving right triangles and to find the lengths of sides and the measures of angles in engineering applications.
Physics: The Pythagorean identity is used to solve problems involving right triangles and to find the lengths of sides and the measures of angles in physics applications.

Q: How can I use the Pythagorean identity to solve problems?

A: To use the Pythagorean identity to solve problems, you can follow these steps:

Identify the problem: Identify the problem you want to solve and determine if it involves a right triangle.
Use the Pythagorean identity: Use the Pythagorean identity to relate the sine and cosine functions to the problem.
Solve for the unknown: Solve for the unknown value using the Pythagorean identity and the given information.
Check your answer: Check your answer to make sure it is correct.

Q: What are some common mistakes to avoid when using the Pythagorean identity?

A: Some common mistakes to avoid when using the Pythagorean identity include:

Not using the correct values for sine and cosine: Make sure to use the correct values for sine and cosine for the given angle.
Not simplifying the equation: Make sure to simplify the equation before solving for the unknown value.
Not checking your answer: Make sure to check your answer to make sure it is correct.

Q: How can I practice using the Pythagorean identity?

A: To practice using the Pythagorean identity, you can try the following:

Work through examples: Work through examples of the Pythagorean identity to practice using it.
Solve problems: Solve problems that involve the Pythagorean identity to practice using it in different contexts.
Take practice quizzes: Take practice quizzes to test your understanding of the Pythagorean identity and to identify areas where you need to improve.

Conclusion

In this article, we answered some frequently asked questions about the Pythagorean identity. The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine functions. It is used to solve problems involving right triangles and is a key concept in calculus and other areas of mathematics. By understanding the Pythagorean identity and how to use it, you can solve a wide range of problems in mathematics and other fields.