Suppose X=\sin ^{-1}\left(\frac{7}{19}\right ].Then X X X Will Be An Angle In Quadrant □ \square □ 1 (list All That Apply).Part 2 Of 4: Sin ⁡ ( X ) = 7 19 \sin (x)=\frac{7}{19} Sin ( X ) = 19 7 ​ Part 3 Of 4:$\cos (x) = $\square$

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. In this article, we will explore how to solve trigonometric equations and find the quadrant of an angle. We will use the given equation $x=\sin ^{-1}\left(\frac{7}{19}\right)$ to find the value of $x$ and determine the quadrant in which it lies.

Understanding the Problem

The given equation is $x=\sin ^{-1}\left(\frac{7}{19}\right)$. This equation represents the inverse sine function, which gives us the angle whose sine is equal to a given value. In this case, the value is $\frac{7}{19}$.

Finding the Value of $x$

To find the value of $x$, we need to take the inverse sine of $\frac{7}{19}$. This can be done using a calculator or by using the inverse sine function on a graphing calculator.

import math
x = math.asin(7/19)
print(x)

Determining the Quadrant of $x$

Once we have found the value of $x$, we need to determine the quadrant in which it lies. The sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants.

import math
x = math.asin(7/19)
quadrant = 1  # First quadrant
if math.cos(x) < 0:
quadrant = 2  # Second quadrant
elif math.cos(x) > 0:
quadrant = 1  # First quadrant
else:
quadrant = 3  # Third quadrant
print("The quadrant of x is:", quadrant)

Finding the Value of $\cos(x)$

To determine the quadrant of $x$, we need to find the value of $\cos(x)$. We can do this using the Pythagorean identity:

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

We know that $\sin(x) = \frac{7}{19}$, so we can substitute this value into the equation:

$$\cos^2(x) + \left(\frac{7}{19}\right)^2 = 1$$

Solving for $\cos(x)$, we get:

$$\cos(x) = \pm \sqrt{1 - \left(\frac{7}{19}\right)^2}$$

Since $\cos(x)$ is positive in the first and fourth quadrants, and negative in the second and third quadrants, we can determine the quadrant of $x$ based on the value of $\cos(x)$.

Conclusion

In this article, we have solved the trigonometric equation $x=\sin ^{-1}\left(\frac{7}{19}\right)$ and determined the quadrant of the angle $x$. We have also found the value of $\cos(x)$ using the Pythagorean identity. The final answer is that the angle $x$ lies in the first quadrant.

Part 2 of 4: $\sin (x)=\frac{7}{19}$

The sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants. Therefore, the value of $\sin(x)$ is positive in the first and second quadrants, and negative in the third and fourth quadrants.

Part 3 of 4: $\cos (x) = \boxed{\frac{12}{19}}$

To find the value of $\cos(x)$, we can use the Pythagorean identity:

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

We know that $\sin(x) = \frac{7}{19}$, so we can substitute this value into the equation:

$$\cos^2(x) + \left(\frac{7}{19}\right)^2 = 1$$

Solving for $\cos(x)$, we get:

$$\cos(x) = \pm \sqrt{1 - \left(\frac{7}{19}\right)^2}$$

Since $\cos(x)$ is positive in the first and fourth quadrants, and negative in the second and third quadrants, we can determine the quadrant of $x$ based on the value of $\cos(x)$.

Part 4 of 4: Quadrant of $x$

Based on the value of $\cos(x)$, we can determine the quadrant of $x$. Since $\cos(x)$ is positive, the angle $x$ lies in the first quadrant.

Conclusion

$$$$$$$$

Q: What is the inverse sine function?

A: The inverse sine function, denoted by $\sin^{-1}(x)$, is a function that gives us the angle whose sine is equal to a given value. In other words, if $\sin(x) = y$, then $\sin^{-1}(y) = x$.

Q: How do we find the value of $x$ in the equation $x=\sin ^{-1}\left(\frac{7}{19}\right)$?

A: To find the value of $x$, we need to take the inverse sine of $\frac{7}{19}$. This can be done using a calculator or by using the inverse sine function on a graphing calculator.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states:

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

This identity can be used to find the value of $\cos(x)$ or $\sin(x)$ if we know the value of the other trigonometric function.

Q: How do we determine the quadrant of an angle?

A: To determine the quadrant of an angle, we need to consider the signs of the trigonometric functions. The sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants. The cosine function is positive in the first and fourth quadrants, and negative in the second and third quadrants.

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

A: The sine and cosine functions are related by the Pythagorean identity:

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

This identity shows that the sine and cosine functions are complementary, meaning that they are equal in magnitude but opposite in sign.

Q: How do we find the value of $\cos(x)$ in the equation $x=\sin ^{-1}\left(\frac{7}{19}\right)$?

A: To find the value of $\cos(x)$, we can use the Pythagorean identity:

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

We know that $\sin(x) = \frac{7}{19}$, so we can substitute this value into the equation:

$$\cos^2(x) + \left(\frac{7}{19}\right)^2 = 1$$

Solving for $\cos(x)$, we get:

$$\cos(x) = \pm \sqrt{1 - \left(\frac{7}{19}\right)^2}$$

Q: What is the final answer to the problem?

A: The final answer to the problem is that the angle $x$ lies in the first quadrant.

Q: What are some common mistakes to avoid when working with trigonometric equations?

A: Some common mistakes to avoid when working with trigonometric equations include:

• Not considering the signs of the trigonometric functions
• Not using the Pythagorean identity to find the value of $\cos(x)$ or $\sin(x)$
• Not checking the quadrant of the angle
• Not using a calculator or graphing calculator to find the value of $x$

Q: How can I practice solving trigonometric equations?

A: You can practice solving trigonometric equations by working through example problems and exercises. You can also use online resources, such as Khan Academy or Wolfram Alpha, to practice solving trigonometric equations. Additionally, you can try solving trigonometric equations on your own by using a calculator or graphing calculator to find the value of $x$.