7. If $\sin (\theta) = \frac{7}{25}$, Find $\cos (\theta$\] Using The Pythagorean Identity.8. Express $\cot(\theta$\] In Terms Of Sine And Cosine.9. Show That $1 + \tan^2(\theta) = \sec^2(\theta$\].10. If $\sin

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will explore some of the key concepts in trigonometry, including the Pythagorean identity, and how they can be used to solve problems involving trigonometric functions.

The Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine of an angle to the tangent of that angle. It is given by the equation:

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

This identity can be used to find the cosine of an angle if the sine of the angle is known, and vice versa. We will use this identity to solve the first problem.

Problem 1: Find $\cos(\theta)$ using the Pythagorean identity

Given that $\sin(\theta) = \frac{7}{25}$, we can use the Pythagorean identity to find the cosine of the angle.

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

Substituting the given value of $\sin(\theta)$, we get:

$\left(\frac{7}{25}\right)^2 + \cos^2(\theta) = 1$

Simplifying the equation, we get:

$\frac{49}{625} + \cos^2(\theta) = 1$

Subtracting $\frac{49}{625}$ from both sides, we get:

$\cos^2(\theta) = \frac{576}{625}$

Taking the square root of both sides, we get:

$\cos(\theta) = \pm \sqrt{\frac{576}{625}}$

Simplifying the expression, we get:

$\cos(\theta) = \pm \frac{24}{25}$

Therefore, the cosine of the angle is either $\frac{24}{25}$ or $-\frac{24}{25}$.

Problem 2: Express $\cot(\theta)$ in terms of sine and cosine

The cotangent of an angle is defined as the ratio of the cosine of the angle to the sine of the angle. We can express $\cot(\theta)$ in terms of sine and cosine using the following equation:

$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Substituting the given value of $\sin(\theta)$, we get:

$\cot(\theta) = \frac{\cos(\theta)}{\frac{7}{25}}$

Simplifying the expression, we get:

$\cot(\theta) = \frac{25\cos(\theta)}{7}$

Therefore, the cotangent of the angle can be expressed in terms of sine and cosine as $\frac{25\cos(\theta)}{7}$.

Problem 3: Show that $1 + \tan^2(\theta) = \sec^2(\theta)$

We can use the Pythagorean identity to show that $1 + \tan^2(\theta) = \sec^2(\theta)$. We start by expressing $\tan(\theta)$ in terms of sine and cosine:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Squaring both sides, we get:

$\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}$

Substituting the Pythagorean identity, we get:

$\tan^2(\theta) = \frac{1 - \cos^2(\theta)}{\cos^2(\theta)}$

Simplifying the expression, we get:

$\tan^2(\theta) = \frac{1}{\cos^2(\theta)} - 1$

Adding 1 to both sides, we get:

$1 + \tan^2(\theta) = \frac{1}{\cos^2(\theta)}$

Defining $\sec(\theta)$ as the reciprocal of $\cos(\theta)$, we get:

$\sec(\theta) = \frac{1}{\cos(\theta)}$

Substituting this definition, we get:

$1 + \tan^2(\theta) = \sec^2(\theta)$

Therefore, we have shown that $1 + \tan^2(\theta) = \sec^2(\theta)$.

Problem 4: If $\sin(\theta) = \frac{3}{5}$, find $\cos(\theta)$ using the Pythagorean identity

Given that $\sin(\theta) = \frac{3}{5}$, we can use the Pythagorean identity to find the cosine of the angle.

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

Substituting the given value of $\sin(\theta)$, we get:

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

Simplifying the equation, we get:

$\frac{9}{25} + \cos^2(\theta) = 1$

Subtracting $\frac{9}{25}$ from both sides, we get:

$\cos^2(\theta) = \frac{16}{25}$

Taking the square root of both sides, we get:

$\cos(\theta) = \pm \sqrt{\frac{16}{25}}$

Simplifying the expression, we get:

$\cos(\theta) = \pm \frac{4}{5}$

Therefore, the cosine of the angle is either $\frac{4}{5}$ or $-\frac{4}{5}$.

Conclusion

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Introduction

In our previous article, we explored some of the key concepts in trigonometry, including the Pythagorean identity and how it can be used to solve problems involving trigonometric functions. In this article, we will answer some of the most frequently asked questions about trigonometric identities and the Pythagorean theorem.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that relates the sine and cosine of an angle to the tangent of that angle. It is given by the equation:

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

Q: How can I use the Pythagorean identity to find the cosine of an angle given the sine of the angle?

A: To find the cosine of an angle given the sine of the angle, you can use the Pythagorean identity as follows:

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

Substitute the given value of $\sin(\theta)$ into the equation and solve for $\cos(\theta)$.

Q: How can I express the cotangent of an angle in terms of sine and cosine?

A: The cotangent of an angle can be expressed in terms of sine and cosine as follows:

$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Q: How can I show that $1 + \tan^2(\theta) = \sec^2(\theta)$?

A: To show that $1 + \tan^2(\theta) = \sec^2(\theta)$, you can use the Pythagorean identity as follows:

$\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}$

Substitute the Pythagorean identity into the equation and simplify to show that $1 + \tan^2(\theta) = \sec^2(\theta)$.

Q: What is the difference between the sine and cosine of an angle?

A: The sine and cosine of an angle are both trigonometric functions that relate the angle to the ratio of the lengths of the sides of a right triangle. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How can I use the Pythagorean identity to find the sine of an angle given the cosine of the angle?

A: To find the sine of an angle given the cosine of the angle, you can use the Pythagorean identity as follows:

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

Substitute the given value of $\cos(\theta)$ into the equation and solve for $\sin(\theta)$.

Q: What is the tangent of an angle?

A: The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. It is given by the equation:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Q: How can I use the Pythagorean identity to find the tangent of an angle given the sine and cosine of the angle?

A: To find the tangent of an angle given the sine and cosine of the angle, you can use the Pythagorean identity as follows:

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

Substitute the given values of $\sin(\theta)$ and $\cos(\theta)$ into the equation and solve for $\tan(\theta)$.

Conclusion

In this article, we have answered some of the most frequently asked questions about trigonometric identities and the Pythagorean theorem. We have covered topics such as the Pythagorean identity, the sine and cosine of an angle, the tangent of an angle, and how to use the Pythagorean identity to solve problems involving trigonometric functions. We hope that this article has been helpful in clarifying some of the key concepts in trigonometry.