Use The Appropriate Identity To Find The Indicated Function Value. Rationalize The Denominator, If Necessary. If The Answer Is A Decimal, Round Your Answer To Three Decimal Places.Find Cos ⁡ Θ \cos \theta Cos Θ , Given That Sec ⁡ Θ = − 7 \sec \theta = -7 Sec Θ = − 7 .A.

Introduction

In trigonometry, we often encounter problems that require us to find the value of a trigonometric function given the value of another function. One of the most useful tools for solving these problems is the Pythagorean identity, which relates the sine, cosine, and tangent functions. In this article, we will explore how to use the Pythagorean identity to find the value of $\cos \theta$ given that $\sec \theta = -7$.

Understanding the Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine, cosine, and tangent functions. It states that for any angle $\theta$,

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

This identity can be used to find the value of one trigonometric function given the value of another function. For example, if we know the value of $\sin \theta$, we can use the Pythagorean identity to find the value of $\cos \theta$.

Finding $\cos \theta$ Using the Pythagorean Identity

Given that $\sec \theta = -7$, we can use the Pythagorean identity to find the value of $\cos \theta$. The secant function is defined as the reciprocal of the cosine function, so we can write

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

Substituting $\sec \theta = -7$ into this equation, we get

$$\frac{1}{\cos \theta} = -7$$

To solve for $\cos \theta$, we can multiply both sides of the equation by $\cos \theta$ to get

$$1 = -7 \cos \theta$$

Now, we can divide both sides of the equation by $-7$ to get

$$\cos \theta = -\frac{1}{7}$$

Therefore, the value of $\cos \theta$ is $-\frac{1}{7}$.

Rationalizing the Denominator

In some cases, we may need to rationalize the denominator of a fraction. Rationalizing the denominator involves multiplying the numerator and denominator by a radical that will eliminate the radical in the denominator. For example, if we have the fraction $\frac{1}{\sqrt{2}}$, we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$ to get

$$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

In the case of the fraction $-\frac{1}{7}$, we do not need to rationalize the denominator because it is already in its simplest form.

Conclusion

In this article, we have seen how to use the Pythagorean identity to find the value of $\cos \theta$ given that $\sec \theta = -7$. We have also seen how to rationalize the denominator of a fraction. By using these tools, we can solve a wide range of trigonometric problems and gain a deeper understanding of the relationships between the different trigonometric functions.

Real-World Applications

The Pythagorean identity has many real-world applications in fields such as physics, engineering, and computer science. For example, in physics, the Pythagorean identity is used to describe the relationships between the sine, cosine, and tangent functions in the context of wave motion. In engineering, the Pythagorean identity is used to design and analyze systems that involve trigonometric functions, such as bridges and buildings. In computer science, the Pythagorean identity is used in algorithms that involve trigonometric functions, such as graphics and game development.

Common Mistakes to Avoid

When using the Pythagorean identity to find the value of a trigonometric function, there are several common mistakes to avoid. One of the most common mistakes is to forget to rationalize the denominator of a fraction. Another common mistake is to use the Pythagorean identity in a way that is not consistent with the given information. For example, if we are given that $\sec \theta = -7$, we should not use the Pythagorean identity to find the value of $\tan \theta$ unless we are also given the value of $\sin \theta$.

Practice Problems

To practice using the Pythagorean identity to find the value of a trigonometric function, try the following problems:

1. Find the value of $\cos \theta$ given that $\sec \theta = 3$.
2. Find the value of $\sin \theta$ given that $\csc \theta = -2$.
3. Find the value of $\tan \theta$ given that $\sec \theta = -5$.

Answer Key

1. $\cos \theta = -\frac{1}{3}$
2. $\sin \theta = \frac{1}{2}$
3. $\tan \theta = \frac{1}{5}$

Conclusion

Q: What is the Pythagorean identity?

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

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

Q: How can I use the Pythagorean identity to find the value of $\cos \theta$ given that $\sec \theta = -7$?

A: To find the value of $\cos \theta$ given that $\sec \theta = -7$, we can use the Pythagorean identity as follows:

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

Substituting $\sec \theta = -7$ into this equation, we get

$$\frac{1}{\cos \theta} = -7$$

To solve for $\cos \theta$, we can multiply both sides of the equation by $\cos \theta$ to get

$$1 = -7 \cos \theta$$

Now, we can divide both sides of the equation by $-7$ to get

$$\cos \theta = -\frac{1}{7}$$

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are two of the most fundamental trigonometric functions. The sine function is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.

Q: How can I use the Pythagorean identity to find the value of $\tan \theta$ given that $\sec \theta = -5$?

A: To find the value of $\tan \theta$ given that $\sec \theta = -5$, we can use the Pythagorean identity as follows:

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

Substituting $\sec \theta = -5$ into this equation, we get

$$\frac{1}{\cos \theta} = -5$$

To solve for $\cos \theta$, we can multiply both sides of the equation by $\cos \theta$ to get

$$1 = -5 \cos \theta$$

Now, we can divide both sides of the equation by $-5$ to get

$$\cos \theta = -\frac{1}{5}$$

We can then use the definition of the tangent function to find the value of $\tan \theta$:

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

Substituting $\cos \theta = -\frac{1}{5}$ into this equation, we get

$$\tan \theta = \frac{\sin \theta}{-\frac{1}{5}}$$

To solve for $\tan \theta$, we can multiply both sides of the equation by $-\frac{1}{5}$ to get

$$\tan \theta = -5 \sin \theta$$

Q: What is the relationship between the sine, cosine, and tangent functions?

A: The sine, cosine, and tangent functions are three of the most fundamental trigonometric functions. The sine function is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse. The tangent function is defined as the ratio of the length of the side opposite a given angle to the length of the side adjacent to a given angle.

Q: How can I use the Pythagorean identity to find the value of $\csc \theta$ given that $\sin \theta = \frac{1}{2}$?

A: To find the value of $\csc \theta$ given that $\sin \theta = \frac{1}{2}$, we can use the Pythagorean identity as follows:

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

Substituting $\sin \theta = \frac{1}{2}$ into this equation, we get

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

Simplifying this equation, we get

$$\frac{1}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{1}{4}$ from both sides of the equation, we get

$$\cos^2 \theta = \frac{3}{4}$$

Taking the square root of both sides of the equation, we get

$$\cos \theta = \pm \frac{\sqrt{3}}{2}$$

We can then use the definition of the cosecant function to find the value of $\csc \theta$:

$$\csc \theta = \frac{1}{\sin \theta}$$

Substituting $\sin \theta = \frac{1}{2}$ into this equation, we get

$$\csc \theta = \frac{1}{\frac{1}{2}}$$

Simplifying this equation, we get

$$\csc \theta = 2$$

Q: What is the relationship between the Pythagorean identity and the trigonometric functions?

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

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

This identity can be used to find the value of one trigonometric function given the value of another function. For example, if we know the value of $\sin \theta$, we can use the Pythagorean identity to find the value of $\cos \theta$.

Q: How can I use the Pythagorean identity to find the value of $\sec \theta$ given that $\cos \theta = -\frac{1}{3}$?

A: To find the value of $\sec \theta$ given that $\cos \theta = -\frac{1}{3}$, we can use the Pythagorean identity as follows:

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

Substituting $\cos \theta = -\frac{1}{3}$ into this equation, we get

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

Simplifying this equation, we get

$$\sin^2 \theta + \frac{1}{9} = 1$$

Subtracting $\frac{1}{9}$ from both sides of the equation, we get

$$\sin^2 \theta = \frac{8}{9}$$

Taking the square root of both sides of the equation, we get

$$\sin \theta = \pm \frac{2\sqrt{2}}{3}$$

We can then use the definition of the secant function to find the value of $\sec \theta$:

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

Substituting $\cos \theta = -\frac{1}{3}$ into this equation, we get

$$\sec \theta = \frac{1}{-\frac{1}{3}}$$

Simplifying this equation, we get

$$\sec \theta = -3$$

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

A: When using the Pythagorean identity, there are several common mistakes to avoid. One of the most common mistakes is to forget to rationalize the denominator of a fraction. Another common mistake is to use the Pythagorean identity in a way that is not consistent with the given information. For example, if we are given that $\sec \theta = -7$, we should not use the Pythagorean identity to find the value of $\tan \theta$ unless we are also given the value of $\sin \theta$.