Fill In The Blank. If Necessary, Use The Slash Mark ( / ) For A Fraction Bar.If Sin ⁡ Θ = 3 5 \sin \theta = \frac{3}{5} Sin Θ = 5 3 ​ , Then Tan ⁡ Θ = \tan \theta = \quad Tan Θ = Answer Here: __________

Solving Trigonometric Equations: Finding the Value of $\tan \theta$

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 finding the value of $\tan \theta$ given the value of $\sin \theta$.

Understanding Trigonometric Functions

Before we dive into solving the equation, let's briefly review the trigonometric functions involved. The sine, cosine, and tangent functions are defined as follows:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Given Information

We are given that $\sin \theta = \frac{3}{5}$. This means that the ratio of the opposite side to the hypotenuse is $\frac{3}{5}$.

Finding the Value of $\tan \theta$

To find the value of $\tan \theta$, we need to find the values of the opposite and adjacent sides. We can use the Pythagorean theorem to find the value of the adjacent side.

Using the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. Mathematically, it can be expressed as:

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

where $a$ and $b$ are the lengths of the two sides, and $c$ is the length of the hypotenuse.

Applying the Pythagorean Theorem

In our case, we know the value of the hypotenuse ($\frac{5}{1}$) and the value of the opposite side ($\frac{3}{1}$). We can use the Pythagorean theorem to find the value of the adjacent side.

$a^2 + b^2 = c^2$ $a^2 + \left(\frac{3}{1}\right)^2 = \left(\frac{5}{1}\right)^2$ $a^2 + \frac{9}{1} = \frac{25}{1}$ $a^2 = \frac{25}{1} - \frac{9}{1}$ $a^2 = \frac{16}{1}$ $a = \sqrt{\frac{16}{1}}$ $a = \frac{4}{1}$

Finding the Value of $\tan \theta$

Now that we have the values of the opposite and adjacent sides, we can find the value of $\tan \theta$.

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ $\tan \theta = \frac{\frac{3}{1}}{\frac{4}{1}}$ $\tan \theta = \frac{3}{4}$

In this article, we solved a trigonometric equation to find the value of $\tan \theta$ given the value of $\sin \theta$. We used the Pythagorean theorem to find the value of the adjacent side and then used the definition of the tangent function to find the value of $\tan \theta$. The final answer is $\boxed{\frac{3}{4}}$.
Frequently Asked Questions: Trigonometric Equations and Identities

In our previous article, we solved a trigonometric equation to find the value of $\tan \theta$ given the value of $\sin \theta$. In this article, we will answer some frequently asked questions related to trigonometric equations and identities.

Q: What is the difference between a trigonometric equation and a trigonometric identity?

A: A trigonometric equation is an equation that involves trigonometric functions, such as $\sin \theta$, $\cos \theta$, and $\tan \theta$. A trigonometric identity, on the other hand, is a statement that is true for all values of the trigonometric functions involved. For example, the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ is a trigonometric identity.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to isolate the trigonometric function on one side of the equation. You can use various techniques, such as factoring, the Pythagorean theorem, and trigonometric identities, to solve the equation.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in trigonometry that states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides. Mathematically, it can be expressed as:

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

where $a$ and $b$ are the lengths of the two sides, and $c$ is the length of the hypotenuse.

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

A: To use the Pythagorean theorem to solve a trigonometric equation, you need to identify the lengths of the two sides and the hypotenuse in the equation. You can then use the Pythagorean theorem to find the value of the unknown side.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$
• $\sec \theta = \frac{1}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$

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

A: To use trigonometric identities to solve a trigonometric equation, you need to identify the trigonometric functions involved in the equation. You can then use the trigonometric identities to simplify the equation and solve for the unknown variable.

Q: What are some common trigonometric equations?

A: Some common trigonometric equations include:

• $\sin \theta = \frac{3}{5}$
• $\cos \theta = \frac{4}{5}$
• $\tan \theta = \frac{3}{4}$
• $\cot \theta = \frac{4}{3}$
• $\sec \theta = \frac{5}{4}$
• $\csc \theta = \frac{5}{3}$

In this article, we answered some frequently asked questions related to trigonometric equations and identities. We covered topics such as the difference between a trigonometric equation and a trigonometric identity, how to solve a trigonometric equation, and common trigonometric identities and equations. We hope that this article has been helpful in clarifying some of the concepts related to trigonometric equations and identities.