# Is tan theta equal to sin theta?

Answered by Willie Powers

Tangent of theta (tan theta) is not equal to sine of theta (sin theta). They are related, but they are not the same. The tangent function is defined as the ratio of the sine function to the cosine function. In other words, tan theta is equal to sin theta divided by cos theta.

To understand this relationship better, let’s take a look at the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is often used in trigonometry to visualize the values of trigonometric functions for different angles.

Let’s consider an angle theta in standard position, where the initial side of the angle coincides with the positive x-axis and the terminal side of the angle intersects the unit circle at a point (x, y). The x-coordinate of this point represents cos theta, and the y-coordinate represents sin theta.

Now, let’s calculate the tangent of theta. The tangent function is defined as the ratio of the sine function to the cosine function, so we have:

Tan theta = sin theta / cos theta

This means that the value of tan theta is determined by the ratio of the y-coordinate to the x-coordinate of the point (x, y) on the unit circle.

It is important to note that the tangent function has certain properties. For example, tangent is positive in the first and third quadrants of the unit circle, where both sine and cosine are positive or negative. However, tangent is negative in the second and fourth quadrants, where sine is positive and cosine is negative, or vice versa.

So, while tangent and sine are related through the trigonometric definition, they are not equal to each other. The tangent function takes into account both the sine and cosine values of an angle, while the sine function only considers the y-coordinate of the point on the unit circle.

Tan theta is not equal to sin theta. Tangent is defined as the ratio of sine to cosine, and its value depends on the coordinates of the point on the unit circle, while sine only considers the y-coordinate.