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

by ADMIN 209 views

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 Quadrants

Before we proceed, it is essential to understand the concept of quadrants in trigonometry. The unit circle is divided into four quadrants, each representing a specific range of angles. The quadrants are:

  • Quadrant I: 0∘≤θ<90∘0^\circ \leq \theta < 90^\circ
  • Quadrant II: 90∘≤θ<180∘90^\circ \leq \theta < 180^\circ
  • Quadrant III: 180∘≤θ<270∘180^\circ \leq \theta < 270^\circ
  • Quadrant IV: 270∘≤θ<360∘270^\circ \leq \theta < 360^\circ

In this article, we are dealing with an angle in quadrant IV, which means that the cosine of the angle is positive, and the sine of the angle is negative.

**The Equation cos⁡(θ)=89\cos (\theta) = \frac{8}{9}

The given equation is cos⁡(θ)=89\cos (\theta) = \frac{8}{9}. To solve this equation, we need to find the value of θ\theta that satisfies this equation. We can start by using the Pythagorean identity, which states that sin⁡2(θ)+cos⁡2(θ)=1\sin^2 (\theta) + \cos^2 (\theta) = 1.

Using the Pythagorean Identity

We can rewrite the Pythagorean identity as sin⁡2(θ)=1−cos⁡2(θ)\sin^2 (\theta) = 1 - \cos^2 (\theta). Substituting the value of cos⁡(θ)\cos (\theta) from the given equation, we get:

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

Simplifying the equation, we get:

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

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

Taking the square root of both sides, we get:

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

Since the angle is in quadrant IV, the sine of the angle is negative. Therefore, we take the negative square root:

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 definition of tangent:

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

Substituting the values of sin⁡(θ)\sin (\theta) and cos⁡(θ)\cos (\theta), we get:

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

Simplifying the equation, we get:

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 definition of tangent to find the value of tan⁡(θ)\tan (\theta). The final answer is −178\boxed{-\frac{\sqrt{17}}{8}}.

Discussion

This problem is a classic example of a trigonometric equation that requires the use of the Pythagorean identity and the definition of tangent. The solution involves a series of algebraic manipulations and trigonometric identities. The final answer is a rational expression involving square roots.

Additional Resources

For more information on trigonometry and solving trigonometric equations, please refer to the following resources:

  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: Trigonometry
  • Wolfram MathWorld: Trigonometry

FAQs

  • What is the value of tan⁡(θ)\tan (\theta) if cos⁡(θ)=89\cos (\theta) = \frac{8}{9} and θ\theta is an angle in quadrant IV?
    • The value of tan⁡(θ)\tan (\theta) is −178-\frac{\sqrt{17}}{8}.
  • How do I solve a trigonometric equation involving the cosine function?
    • You can use the Pythagorean identity to find the value of the sine function and then use the definition of tangent to find the value of the tangent function.
  • What is the definition of tangent in trigonometry?
    • The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle.
      Trigonometry Q&A: Frequently Asked Questions =====================================================

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, covering topics such as solving trigonometric equations, understanding the quadrants, and using trigonometric identities.

Q: What is the value of tan⁡(θ)\tan (\theta) if cos⁡(θ)=89\cos (\theta) = \frac{8}{9} and θ\theta is an angle in quadrant IV?

A: The value of tan⁡(θ)\tan (\theta) is −178-\frac{\sqrt{17}}{8}.

Q: How do I solve a trigonometric equation involving the cosine function?

A: You can use the Pythagorean identity to find the value of the sine function and then use the definition of tangent to find the value of the tangent function.

Q: What is the definition of tangent in trigonometry?

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

Q: What is the Pythagorean identity in trigonometry?

A: The Pythagorean identity is a fundamental identity in trigonometry that states that sin⁡2(θ)+cos⁡2(θ)=1\sin^2 (\theta) + \cos^2 (\theta) = 1.

Q: How do I use the Pythagorean identity to solve a trigonometric equation?

A: You can use the Pythagorean identity to find the value of the sine function by rearranging the equation to isolate the sine function. For example, if you have the equation cos⁡(θ)=89\cos (\theta) = \frac{8}{9}, you can use the Pythagorean identity to find the value of the sine function.

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

A: The sine and cosine functions are two fundamental functions in trigonometry that describe the relationships between the sides and angles of triangles. The sine function is defined as the ratio of the opposite side to the hypotenuse, while the cosine function is defined as the ratio of the adjacent side to the hypotenuse.

Q: How do I use trigonometric identities to solve a trigonometric equation?

A: You can use trigonometric identities to simplify and solve trigonometric equations. For example, you can use the Pythagorean identity to find the value of the sine function and then use the definition of tangent to find the value of the tangent function.

Q: What is the significance of the quadrants in trigonometry?

A: The quadrants are a fundamental concept in trigonometry that describe the relationships between the angles and sides of triangles. The quadrants are used to determine the signs of the sine and cosine functions for different angles.

Q: How do I determine the signs of the sine and cosine functions for different angles?

A: You can use the quadrants to determine the signs of the sine and cosine functions for different angles. For example, if an angle is in quadrant I, the sine function is positive, and the cosine function is positive.

Conclusion

In this article, we have answered some frequently asked questions about trigonometry, covering topics such as solving trigonometric equations, understanding the quadrants, and using trigonometric identities. We hope that this article has provided you with a better understanding of trigonometry and its applications.

Additional Resources

For more information on trigonometry and solving trigonometric equations, please refer to the following resources:

  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: Trigonometry
  • Wolfram MathWorld: Trigonometry

FAQs

  • What is the value of tan⁡(θ)\tan (\theta) if cos⁡(θ)=89\cos (\theta) = \frac{8}{9} and θ\theta is an angle in quadrant IV?
    • The value of tan⁡(θ)\tan (\theta) is −178-\frac{\sqrt{17}}{8}.
  • How do I solve a trigonometric equation involving the cosine function?
    • You can use the Pythagorean identity to find the value of the sine function and then use the definition of tangent to find the value of the tangent function.
  • What is the definition of tangent in trigonometry?
    • The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle.