Given Tan ⁡ Θ = − 3 4 \tan \theta = \frac{-3}{4} Tan Θ = 4 − 3 ​ Where Θ \theta Θ Is In Quadrant IV, Match The Half-angle Trigonometric Expression With The Corresponding Exact Value.Half-Angle Trigonometric Expression:A. \cos \left(\frac{\theta}{2}\right ]

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the half-angle trigonometric expression, which is used to find the values of trigonometric functions for half of an angle. In this article, we will explore the half-angle trigonometric expression and match it with the corresponding exact value for a given angle.

What is Half-Angle Trigonometric Expression?

The half-angle trigonometric expression is a formula used to find the values of trigonometric functions for half of an angle. It is defined as:

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

$$\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$

$$\tan \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$$

Given Information

We are given that $\tan \theta = \frac{-3}{4}$, where $\theta$ is in Quadrant IV. This means that the angle $\theta$ is in the fourth quadrant, where the cosine function is positive and the sine function is negative.

Finding the Exact Value

To find the exact value of the half-angle trigonometric expression, we need to find the value of $\cos \theta$. We can use the given information to find the value of $\cos \theta$.

Since $\tan \theta = \frac{-3}{4}$, we can draw a right triangle with the opposite side as -3 and the adjacent side as 4. The hypotenuse of the triangle can be found using the Pythagorean theorem:

$$c^2 = a^2 + b^2$$

$$c^2 = (-3)^2 + 4^2$$

$$c^2 = 9 + 16$$

$$c^2 = 25$$

$$c = \sqrt{25}$$

$$c = 5$$

Now that we have the value of the hypotenuse, we can find the value of $\cos \theta$:

$$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$

$$\cos \theta = \frac{4}{5}$$

Matching the Half-Angle Trigonometric Expression

Now that we have the value of $\cos \theta$, we can match the half-angle trigonometric expression with the corresponding exact value.

Using the formula for $\cos \left(\frac{\theta}{2}\right)$, we get:

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

$$\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \frac{4}{5}}{2}}$$

$$\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{\frac{9}{5}}{2}}$$

$$\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{9}{10}}$$

Since $\theta$ is in Quadrant IV, the cosine function is positive, so we take the positive value:

$$\cos \left(\frac{\theta}{2}\right) = \sqrt{\frac{9}{10}}$$

Conclusion

In this article, we explored the half-angle trigonometric expression and matched it with the corresponding exact value for a given angle. We used the given information to find the value of $\cos \theta$ and then used the formula for $\cos \left(\frac{\theta}{2}\right)$ to find the exact value of the half-angle trigonometric expression. The final answer is $\boxed{\sqrt{\frac{9}{10}}}$.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Half-Angle Trigonometric Expressions" by Math Open Reference

Note

Q&A: Half-Angle Trigonometric Expression

Q: What is the half-angle trigonometric expression?

A: The half-angle trigonometric expression is a formula used to find the values of trigonometric functions for half of an angle. It is defined as:

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

$$\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$

$$\tan \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$$

Q: How do I use the half-angle trigonometric expression?

A: To use the half-angle trigonometric expression, you need to know the value of $\cos \theta$. You can find the value of $\cos \theta$ using the given information, such as the value of $\tan \theta$ or the value of the opposite and adjacent sides of a right triangle.

Q: What is the difference between the half-angle trigonometric expression and the double-angle trigonometric expression?

A: The half-angle trigonometric expression is used to find the values of trigonometric functions for half of an angle, while the double-angle trigonometric expression is used to find the values of trigonometric functions for twice an angle.

Q: Can I use the half-angle trigonometric expression to find the values of trigonometric functions for any angle?

A: No, the half-angle trigonometric expression can only be used to find the values of trigonometric functions for angles that are in the first or second quadrant. If the angle is in the third or fourth quadrant, you need to use the double-angle trigonometric expression.

Q: How do I know which value to use for the half-angle trigonometric expression?

A: To determine which value to use for the half-angle trigonometric expression, you need to know the quadrant of the angle. If the angle is in the first or second quadrant, you use the positive value. If the angle is in the third or fourth quadrant, you use the negative value.

Q: Can I use the half-angle trigonometric expression to find the values of trigonometric functions for a specific angle?

A: Yes, you can use the half-angle trigonometric expression to find the values of trigonometric functions for a specific angle. You need to know the value of $\cos \theta$ and then use the formula for the half-angle trigonometric expression.

Q: What are some common applications of the half-angle trigonometric expression?

A: The half-angle trigonometric expression has many applications in mathematics and physics, such as:

• Finding the values of trigonometric functions for half of an angle
• Solving trigonometric equations
• Finding the values of trigonometric functions for specific angles
• Calculating the area and perimeter of triangles

Q: Can I use the half-angle trigonometric expression to find the values of trigonometric functions for a complex angle?

A: Yes, you can use the half-angle trigonometric expression to find the values of trigonometric functions for a complex angle. You need to know the value of $\cos \theta$ and then use the formula for the half-angle trigonometric expression.

Conclusion

In this article, we explored the half-angle trigonometric expression and answered some common questions about it. We discussed how to use the half-angle trigonometric expression, the difference between the half-angle trigonometric expression and the double-angle trigonometric expression, and some common applications of the half-angle trigonometric expression. We also discussed how to determine which value to use for the half-angle trigonometric expression and how to use it to find the values of trigonometric functions for a specific angle.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Half-Angle Trigonometric Expressions" by Math Open Reference

Note

The half-angle trigonometric expression is a powerful tool in trigonometry, and it has many applications in mathematics and physics. In this article, we only explored some common questions and applications of the half-angle trigonometric expression, but there are many other examples and applications that can be explored.