Determine The Tangent And Cotangent Of $\theta$ Given The Following. Round Your Answer To Three Decimal Places, If Necessary.$\sin (\theta) = 0.766$ And $\cos (\theta) = 0.6428$

Determine the Tangent and Cotangent of $\theta$

In trigonometry, the tangent and cotangent of an angle are two fundamental ratios that are used to describe the relationship between the sides of a right-angled triangle. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle, while the cotangent of an angle is the reciprocal of the tangent. In this article, we will determine the tangent and cotangent of an angle $\theta$ given the values of $\sin (\theta)$ and $\cos (\theta)$.

Recall the Definitions of Tangent and Cotangent

Before we proceed, let's recall the definitions of tangent and cotangent.

• Tangent: The tangent of an angle $\theta$ is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. It is denoted by $\tan (\theta)$ and is calculated as $\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}$.
• Cotangent: The cotangent of an angle $\theta$ is the reciprocal of the tangent. It is denoted by $\cot (\theta)$ and is calculated as $\cot (\theta) = \frac{1}{\tan (\theta)} = \frac{\cos (\theta)}{\sin (\theta)}$.

Given Values

We are given the values of $\sin (\theta)$ and $\cos (\theta)$ as follows:

• $\sin (\theta) = 0.766$
• $\cos (\theta) = 0.6428$

Determine the Tangent of $\theta$

To determine the tangent of $\theta$, we can use the formula $\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}$. Plugging in the given values, we get:

$\tan (\theta) = \frac{0.766}{0.6428} = 1.191$

Determine the Cotangent of $\theta$

To determine the cotangent of $\theta$, we can use the formula $\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}$. Plugging in the given values, we get:

$\cot (\theta) = \frac{0.6428}{0.766} = 0.838$

Rounding the Answers

Since we are asked to round our answers to three decimal places, if necessary, we can round the values of $\tan (\theta)$ and $\cot (\theta)$ as follows:

• $\tan (\theta) = 1.191 \approx 1.191$
• $\cot (\theta) = 0.838 \approx 0.838$

In this article, we have determined the tangent and cotangent of an angle $\theta$ given the values of $\sin (\theta)$ and $\cos (\theta)$. We have used the formulas $\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}$ and $\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}$ to calculate the values of $\tan (\theta)$ and $\cot (\theta)$. We have also rounded our answers to three decimal places, if necessary.
Frequently Asked Questions (FAQs) about Determining the Tangent and Cotangent of $\theta$

Q: What is the tangent of an angle?

A: The tangent of an angle $\theta$ is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. It is denoted by $\tan (\theta)$ and is calculated as $\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}$.

Q: What is the cotangent of an angle?

A: The cotangent of an angle $\theta$ is the reciprocal of the tangent. It is denoted by $\cot (\theta)$ and is calculated as $\cot (\theta) = \frac{1}{\tan (\theta)} = \frac{\cos (\theta)}{\sin (\theta)}$.

Q: How do I determine the tangent and cotangent of an angle given the values of $\sin (\theta)$ and $\cos (\theta)$?

A: To determine the tangent and cotangent of an angle $\theta$ given the values of $\sin (\theta)$ and $\cos (\theta)$, you can use the formulas $\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}$ and $\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}$. Simply plug in the given values and calculate the results.

Q: What if I am given the value of $\tan (\theta)$ and I need to find the value of $\sin (\theta)$ and $\cos (\theta)$?

A: If you are given the value of $\tan (\theta)$ and you need to find the value of $\sin (\theta)$ and $\cos (\theta)$, you can use the formulas $\sin (\theta) = \tan (\theta) \cos (\theta)$ and $\cos (\theta) = \frac{1}{\sqrt{1 + \tan^2 (\theta)}}$. Simply plug in the given value of $\tan (\theta)$ and calculate the results.

Q: What if I am given the value of $\cot (\theta)$ and I need to find the value of $\sin (\theta)$ and $\cos (\theta)$?

A: If you are given the value of $\cot (\theta)$ and you need to find the value of $\sin (\theta)$ and $\cos (\theta)$, you can use the formulas $\sin (\theta) = \frac{1}{\sqrt{1 + \cot^2 (\theta)}}$ and $\cos (\theta) = \cot (\theta) \sin (\theta)$. Simply plug in the given value of $\cot (\theta)$ and calculate the results.

Q: Can I use a calculator to determine the tangent and cotangent of an angle?

A: Yes, you can use a calculator to determine the tangent and cotangent of an angle. Most calculators have a built-in function for calculating the tangent and cotangent of an angle. Simply enter the value of the angle and the calculator will give you the result.

Q: What if I am given the value of $\sin (\theta)$ and $\cos (\theta)$ and I need to find the value of $\tan (\theta)$ and $\cot (\theta)$, but the values are not exact?

A: If you are given the value of $\sin (\theta)$ and $\cos (\theta)$ and you need to find the value of $\tan (\theta)$ and $\cot (\theta)$, but the values are not exact, you can use the formulas $\tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)}$ and $\cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)}$. Simply plug in the given values and calculate the results. If the values are not exact, you may need to round the results to a certain number of decimal places.

Q: Can I use a graphing calculator to visualize the tangent and cotangent functions?

A: Yes, you can use a graphing calculator to visualize the tangent and cotangent functions. Simply enter the functions into the calculator and it will graph the functions for you. This can be a useful tool for understanding the behavior of the tangent and cotangent functions.