Consider This Equation: $\cos(\theta) = \frac{8}{9}$If $\theta$ Is An Angle In Quadrant IV, What Is The Value Of $\tan(\theta$\]?A. $\frac{\sqrt{17}}{9}$ B. $-\frac{\sqrt{17}}{8}$ C.

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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 focus on solving trigonometric equations, specifically the equation cos⁡(θ)=89\cos(\theta) = \frac{8}{9}, where θ\theta is an angle in quadrant IV. We will use this equation to find the value of tan⁡(θ)\tan(\theta).

Understanding the Problem

The given equation is cos⁡(θ)=89\cos(\theta) = \frac{8}{9}. This equation represents a relationship between the cosine of an angle θ\theta and a specific value. To solve this equation, we need to find the value of θ\theta that satisfies the equation.

Recalling Trigonometric Identities

Before we proceed, let's recall some basic trigonometric identities that will be useful in solving this equation.

  • cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1
  • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Finding the Value of sin⁡(θ)\sin(\theta)

Using the Pythagorean identity, we can find the value of sin⁡(θ)\sin(\theta).

cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1

sin⁡2(θ)=1−cos⁡2(θ)\sin^2(\theta) = 1 - \cos^2(\theta)

sin⁡2(θ)=1−(89)2\sin^2(\theta) = 1 - \left(\frac{8}{9}\right)^2

sin⁡2(θ)=1−6481\sin^2(\theta) = 1 - \frac{64}{81}

sin⁡2(θ)=1781\sin^2(\theta) = \frac{17}{81}

sin⁡(θ)=±1781\sin(\theta) = \pm \sqrt{\frac{17}{81}}

Since θ\theta is an angle in quadrant IV, the sine of θ\theta is negative.

sin⁡(θ)=−1781\sin(\theta) = -\sqrt{\frac{17}{81}}

Finding the Value of tan⁡(θ)\tan(\theta)

Now that we have the value of sin⁡(θ)\sin(\theta), we can find the value of tan⁡(θ)\tan(\theta) using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.

tan⁡(θ)=−178189\tan(\theta) = \frac{-\sqrt{\frac{17}{81}}}{\frac{8}{9}}

tan⁡(θ)=−178\tan(\theta) = -\frac{\sqrt{17}}{8}

Conclusion

In this article, we solved the trigonometric equation cos⁡(θ)=89\cos(\theta) = \frac{8}{9}, where θ\theta is an angle in quadrant IV. We used the Pythagorean identity to find the value of sin⁡(θ)\sin(\theta) and then used the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} to find the value of tan⁡(θ)\tan(\theta). The final answer is −178\boxed{-\frac{\sqrt{17}}{8}}.

Additional Tips and Tricks

  • When solving trigonometric equations, it's essential to recall the basic trigonometric identities and use them to simplify the equation.
  • When dealing with angles in different quadrants, remember that the sine and cosine functions have different signs in different quadrants.
  • When finding the value of tan⁡(θ)\tan(\theta), use the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} and make sure to simplify the expression.

Common Mistakes to Avoid

  • When solving trigonometric equations, avoid making mistakes in the signs of the sine and cosine functions.
  • When finding the value of tan⁡(θ)\tan(\theta), avoid making mistakes in the simplification of the expression.
  • When dealing with angles in different quadrants, avoid making mistakes in the signs of the sine and cosine functions.

Real-World Applications

Trigonometry has numerous real-world applications, including:

  • Navigation: Trigonometry is used in navigation to calculate distances and directions.
  • Physics: Trigonometry is used in physics to describe the motion of objects.
  • Engineering: Trigonometry is used in engineering to design and build structures.
  • Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

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 answer some frequently asked questions about trigonometry.

Q: What is trigonometry?

A: 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.

Q: What are the basic trigonometric functions?

A: The basic trigonometric functions are:

  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)

Q: What is the Pythagorean identity?

A: The Pythagorean identity is:

cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1

Q: How do I find the value of tan⁡(θ)\tan(\theta)?

A: To find the value of tan⁡(θ)\tan(\theta), you can use the identity:

tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Q: What is the difference between sine and cosine?

A: The sine and cosine functions have different signs in different quadrants. In quadrant I, both sine and cosine are positive. In quadrant II, sine is positive and cosine is negative. In quadrant III, both sine and cosine are negative. In quadrant IV, sine is negative and cosine is positive.

Q: How do I find the value of sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta) for a given angle θ\theta?

A: To find the value of sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta) for a given angle θ\theta, you can use the following formulas:

  • sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{opposite}{hypotenuse}
  • cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{adjacent}{hypotenuse}

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define the trigonometric functions.

Q: How do I use the unit circle to find the value of sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta)?

A: To use the unit circle to find the value of sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta), you can follow these steps:

  1. Draw a point on the unit circle corresponding to the angle θ\theta.
  2. Draw a line from the origin to the point.
  3. The length of the line is the value of sin⁡(θ)\sin(\theta).
  4. The length of the adjacent side is the value of cos⁡(θ)\cos(\theta).

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1
  • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
  • sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}
  • csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}

Q: How do I use trigonometry in real-world applications?

A: Trigonometry has numerous real-world applications, including:

  • Navigation: Trigonometry is used in navigation to calculate distances and directions.
  • Physics: Trigonometry is used in physics to describe the motion of objects.
  • Engineering: Trigonometry is used in engineering to design and build structures.
  • Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. By understanding the basic trigonometric functions, identities, and formulas, you can use trigonometry to solve problems in navigation, physics, engineering, and computer science.