Where is tangent 0 on the unit circle

It removes the need for memorizing different values and allows the user to simply derive different results for different cases. Let's learn more about it and test our understanding with a handy trigonometric calculator I created at the end of the article. The unit circle is a circle with a radius of one unit with its center placed at the origin. In other words, the center is put on a graph where the X and Y axes cross.

Having a radius equal 1 unit will allow us to create reference triangles with hypotenuse equal to 1 unit. As we will see shortly, that allows us to measure sine , cosine and tangent directly.

The triangle below reminds us how we define sine and cosine for some angle alpha. Since the hypotenuse equals 1 and anything divided by 1 equals itself, sin of alpha equals the length of BC. Next, let's move this triangle into our Unit Circle, so the radius of the circle can serve as the hypotenuse. Thus, by moving around the circle and changing the angle, we can measure sine and cosine of that angle by measuring the y and x coordinates accordingly. The point with coordinates 1, 0 corresponds with 0 degrees see Fig 1.

The measure increases in a counterclockwise direction, so the point with coordinates 0, 1 will correspond with 90 degrees. A complete circle — degrees. Next let's see what happens at 90 degrees. The coordinates of the corresponding point are 0, 1.

The circle will look like this:. What about tangent 90? As the cosine measure approaches 0, and it happens to be a denominator in a fraction, the value of that fraction increases to infinity. Therefore tan 90 is said to be undefined. Now the question you might ask: as sin goes from 0 to 1 while cosine goes from 1 to 0, do they ever equal each other? The answer is yes, and that happens exactly half way at 45 degrees!

The circle looks like this:. Finally, the general reference Unit Circle. It reflects both positive and negative values for X and Y axes and shows important values you should remember. As a useful practice tool, I have added a simple trigonometric calculator. It takes inputs for angle measures and outputs corresponding values for sine , cosine and tangent functions.

You can choose degrees or radians as a measure of angle. They each have their advantages and disadvantages. It can be calculated with any desired accuracy. The code for the calculator contains some basic interactivity and error handling within constraints of the editor. Its building blocks are marked and commented so anyone with the desire to modify it can easily do so.

Sign rules for trigonometric functions: The trigonometric functions are each listed in the quadrants in which they are positive. Identifying reference angles will help us identify a pattern in these values.

For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. You will then identify and apply the appropriate sign for that trigonometric function in that quadrant. Since tangent functions are derived from sine and cosine, the tangent can be calculated for any of the special angles by first finding the values for sine or cosine.

However, the rules described above tell us that the sine of an angle in the third quadrant is negative. So we have. The functions sine and cosine can be graphed using values from the unit circle, and certain characteristics can be observed in both graphs. So what do they look like on a graph on a coordinate plane? We can create a table of values and use them to sketch a graph. Again, we can create a table of values and use them to sketch a graph. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers.

A periodic function is a function with a repeated set of values at regular intervals. The diagram below shows several periods of the sine and cosine functions. As we can see in the graph of the sine function, it is symmetric about the origin, which indicates that it is an odd function.

This is characteristic of an odd function: two inputs that are opposites have outputs that are also opposites. Odd symmetry of the sine function: The sine function is odd, meaning it is symmetric about the origin. The graph of the cosine function shows that it is symmetric about the y -axis. This indicates that it is an even function. The shape of the function can be created by finding the values of the tangent at special angles.

However, it is not possible to find the tangent functions for these special angles with the unit circle. We can analyze the graphical behavior of the tangent function by looking at values for some of the special angles. The above points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. At values where the tangent function is undefined, there are discontinuities in its graph. At these values, the graph of the tangent has vertical asymptotes.

As with the sine and cosine functions, tangent is a periodic function. This means that its values repeat at regular intervals.

If we look at any larger interval, we will see that the characteristics of the graph repeat. The graph of the tangent function is symmetric around the origin, and thus is an odd function. Calculate values for the trigonometric functions that are the reciprocals of sine, cosine, and tangent. We have discussed three trigonometric functions: sine, cosine, and tangent.

Each of these functions has a reciprocal function, which is defined by the reciprocal of the ratio for the original trigonometric function. Note that reciprocal functions differ from inverse functions. Inverse functions are a way of working backwards, or determining an angle given a trigonometric ratio; they involve working with the same ratios as the original function. It can be described as the ratio of the length of the hypotenuse to the length of the adjacent side in a triangle.

It is easy to calculate secant with values in the unit circle. Therefore, the secant function for that angle is. It can be described as the ratio of the length of the hypotenuse to the length of the opposite side in a triangle. As with secant, cosecant can be calculated with values in the unit circle.

Therefore, the cosecant function for the same angle is. It can be described as the ratio of the length of the adjacent side to the length of the hypotenuse in a triangle. Cotangent can also be calculated with values in the unit circle. We now recognize six trigonometric functions that can be calculated using values in the unit circle.

Recall that we used values for the sine and cosine functions to calculate the tangent function. We will follow a similar process for the reciprocal functions, referencing the values in the unit circle for our calculations.

In other words:. Describe the characteristics of the graphs of the inverse trigonometric functions, noting their domain and range restrictions. Inverse trigonometric functions are used to find angles of a triangle if we are given the lengths of the sides. Inverse trigonometric functions can be used to determine what angle would yield a specific sine, cosine, or tangent value.

Note that the domain of the inverse function is the range of the original function, and vice versa. However, the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would fail the horizontal line test.

In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods. As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.

These choices for the restricted domains are somewhat arbitrary, but they have important, helpful characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-to-one function that is invertible.

The conventional choice for the restricted domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next, instead of being divided into pieces by an asymptote. We can define the inverse trigonometric functions as follows. Note the domain and range of each function. To find the domain and range of inverse trigonometric functions, we switch the domain and range of the original functions.

Privacy Policy. Skip to main content. Search for:. Trigonometric Functions and the Unit Circle. Learning Objectives Explain the definition of radians in terms of arc length of a unit circle and use this to convert between degrees and radians.

Key Takeaways Key Points One radian is the measure of the central angle of a circle such that the length of the arc is equal to the radius of the circle. Key Terms arc : A continuous part of the circumference of a circle.