Understanding Sine Given Tangent: A Comprehensive Guide

When working with trigonometric functions, it's often necessary to find the sine of an angle given the tangent of that angle. This guide will walk through the process with a detailed example and explore various scenarios.

Introduction to Trigonometric Functions

In trigonometry, three primary functions are used to define the relationships between the angles and sides of a right triangle: sine, cosine, and tangent. These functions are:

Sine (sin): The ratio of the length of the opposite side to the hypotenuse. Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse. Tangent (tan): The ratio of the length of the opposite side to the adjacent side.

Example: Finding Sine Given Tangent

Let's consider a specific example where we are given that tan(theta) frac{3}{4}. We'll use this information to find sin(theta).

Step-by-Step Process

We start with the definition of tangent:

tan(theta) frac{sin(theta)}{cos(theta)}

Given tan(theta) frac{3}{4}, we can write:

sin(theta) 3k cos(theta) 4k

for some positive value of k. The Pythagorean identity states:

sin^2(theta) cos^2(theta) 1

Substituting the expressions for sin(theta) and cos(theta) into the identity:

(3k)^2 (4k)^2 1

This simplifies to:

9k^2 16k^2 1 25k^2 1

Solving for k^2:

k^2 frac{1}{25}

Taking the square root:

k frac{1}{5}

Now we can find sin(theta):

sin(theta) 3k 3 cdot frac{1}{5} frac{3}{5}

Thus, the value of sin(theta) is:

sin(theta) frac{3}{5}

Alternative Methods

Another method to solve this problem involves using the Pythagorean identity with the tangent and secant relationship:

tan^2(theta) sec^2(theta) - 1

Given tan(theta) frac{3}{4}, we find:

sec^2(theta) left(frac{3}{4}right)^2 1 frac{9}{16} 1 frac{25}{16}

So:

cos^2(theta) frac{16}{25} 1 - sin^2(theta) frac{16}{25} sin^2(theta) 1 - frac{16}{25} frac{9}{25}

Thus:

sin(theta) pm frac{3}{5}

Understanding the Trigonometric Identity in Context

In the case where tan(theta) frac{4}{3}, you can visualize it as a 3-4-5 right triangle. Here, the opposite side is 4 units, the adjacent side is 3 units, and the hypotenuse is 5 units. Hence:

sin(theta) frac{opposite}{hypotenuse} frac{4}{5}

However, since the tangent ratio can indicate angles in quadrants 1 or 3, both sin(theta) frac{4}{5} and sin(theta) -frac{4}{5} are valid solutions.

Creating a 3-4-5 Triangle

Given tan(theta) frac{3}{4}, it means that in a right triangle, the opposite side is 3 units, and the adjacent side is 4 units. Using the Pythagorean theorem:

a^2 b^2 c^2

We can calculate the hypotenuse:

4^2 3^2 16 9 25 c sqrt{25} 5

Thus:

sin(theta) frac{opposite}{hypotenuse} frac{3}{5}

Conclusion

By understanding the relationship between sine and tangent, you can solve for the sine of an angle given the tangent. This process involves using trigonometric identities and the Pythagorean theorem. Remember to consider the quadrant of the angle, as this can affect the sign of the sine value.

Related Terms and Keywords

The following are some related terms and keywords that you might find useful when researching or discussing trigonometric functions:

tangent sine trigonometry right triangle Pythagorean identity

These terms can help you find more detailed information and related content on the topic of trigonometric functions.