If Tan ⁡ Θ = 12 5 \tan \theta = \frac{12}{5} Tan Θ = 5 12 ​ , Then Cos ⁡ Θ = \cos \theta = Cos Θ =

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Introduction

In trigonometry, the tangent, cosine, and sine functions are used to describe the relationships between the angles and side lengths of triangles. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Given the value of the tangent function, we can use trigonometric identities to find the values of the cosine and sine functions. In this article, we will explore how to find the value of the cosine function when the tangent function is given.

Understanding the Tangent Function

The tangent function is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Mathematically, it can be represented as:

$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

Given that $\tan \theta = \frac{12}{5}$, we can represent the opposite side as 12 and the adjacent side as 5.

Using Trigonometric Identities to Find Cosine

To find the value of the cosine function, we can use the trigonometric identity:

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

We can also use the identity:

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

Rearranging the second identity, we get:

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

Finding the Value of Sine

To find the value of the sine function, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the side opposite the angle, and the other two sides are the opposite and adjacent sides.

Using the Pythagorean theorem, we can find the length of the hypotenuse:

$$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$$

$$\text{hypotenuse}^2 = 12^2 + 5^2$$

$$\text{hypotenuse}^2 = 144 + 25$$

$$\text{hypotenuse}^2 = 169$$

$$\text{hypotenuse} = \sqrt{169}$$

$$\text{hypotenuse} = 13$$

Finding the Value of Sine

Now that we have the length of the hypotenuse, we can find the value of the sine function:

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

$$\sin \theta = \frac{12}{13}$$

Finding the Value of Cosine

Now that we have the value of the sine function, we can find the value of the cosine function using the identity:

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

$$\cos \theta = \frac{\frac{12}{13}}{\frac{12}{5}}$$

$$\cos \theta = \frac{12}{13} \times \frac{5}{12}$$

$$\cos \theta = \frac{5}{13}$$

Conclusion

In this article, we explored how to find the value of the cosine function when the tangent function is given. We used trigonometric identities and the Pythagorean theorem to find the values of the sine and cosine functions. We found that the value of the cosine function is $\frac{5}{13}$.

Final Answer

The final answer is $\boxed{\frac{5}{13}}$.

Additional Resources

For more information on trigonometry and the tangent, cosine, and sine functions, please refer to the following resources:

• Trigonometry Wikipedia Article
• Tangent, Cosine, and Sine Functions
• Pythagorean Theorem

Related Questions

• What is the value of the sine function when the tangent function is given?
• How can we use the Pythagorean theorem to find the length of the hypotenuse?
• What is the relationship between the tangent, cosine, and sine functions?

Tags

• Trigonometry
• Tangent Function
• Cosine Function
• Sine Function
• Pythagorean Theorem
• Right-Angled Triangle
  

Introduction

In our previous article, we explored how to find the value of the cosine function when the tangent function is given. In this article, we will answer some frequently asked questions about trigonometry and the tangent, cosine, and sine functions.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. It can be represented mathematically as:

$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

Q: What is the cosine function?

A: The cosine function is a trigonometric function that is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It can be represented mathematically as:

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

Q: What is the sine function?

A: The sine function is a trigonometric function that is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle. It can be represented mathematically as:

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

Q: How can we use the Pythagorean theorem to find the length of the hypotenuse?

A: The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In a right-angled triangle, the Pythagorean theorem can be represented mathematically as:

$$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$$

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

A: The tangent, cosine, and sine functions are related to each other through the following identities:

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

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

$$\sin \theta = \tan \theta \cos \theta$$

Q: How can we use the tangent function to find the value of the cosine function?

A: We can use the identity:

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

to find the value of the cosine function when the tangent function is given.

Q: How can we use the cosine function to find the value of the sine function?

A: We can use the identity:

$$\sin \theta = \tan \theta \cos \theta$$

to find the value of the sine function when the cosine function is given.

Q: What is the value of the sine function when the tangent function is given?

A: We can use the identity:

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

to find the value of the sine function when the tangent function is given.

Q: What is the value of the cosine function when the sine function is given?

A: We can use the identity:

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

to find the value of the cosine function when the sine function is given.

Q: How can we use the Pythagorean theorem to find the length of the opposite side?

A: We can use the Pythagorean theorem to find the length of the opposite side by rearranging the equation:

$$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$$

to solve for the opposite side.

Q: How can we use the Pythagorean theorem to find the length of the adjacent side?

A: We can use the Pythagorean theorem to find the length of the adjacent side by rearranging the equation:

$$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$$

to solve for the adjacent side.

Q: What is the relationship between the tangent, cosine, and sine functions in a right-angled triangle?

A: The tangent, cosine, and sine functions are related to each other through the following identities:

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

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

$$\sin \theta = \tan \theta \cos \theta$$

Q: How can we use the tangent function to find the value of the cosine function in a right-angled triangle?

A: We can use the identity:

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

to find the value of the cosine function when the tangent function is given.

Q: How can we use the cosine function to find the value of the sine function in a right-angled triangle?

A: We can use the identity:

$$\sin \theta = \tan \theta \cos \theta$$

to find the value of the sine function when the cosine function is given.

Q: What is the value of the sine function when the tangent function is given in a right-angled triangle?

A: We can use the identity:

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

to find the value of the sine function when the tangent function is given.

Q: What is the value of the cosine function when the sine function is given in a right-angled triangle?

A: We can use the identity:

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

to find the value of the cosine function when the sine function is given.

Conclusion

In this article, we have answered some frequently asked questions about trigonometry and the tangent, cosine, and sine functions. We have explored the relationships between the tangent, cosine, and sine functions and how to use them to find the values of the other functions.

Final Answer

The final answer is $\boxed{\frac{5}{13}}$.

Additional Resources

For more information on trigonometry and the tangent, cosine, and sine functions, please refer to the following resources:

• Trigonometry Wikipedia Article
• Tangent, Cosine, and Sine Functions
• Pythagorean Theorem

Related Questions

• What is the value of the sine function when the tangent function is given?
• How can we use the Pythagorean theorem to find the length of the hypotenuse?
• What is the relationship between the tangent, cosine, and sine functions?

Tags

• Trigonometry
• Tangent Function
• Cosine Function
• Sine Function
• Pythagorean Theorem
• Right-Angled Triangle