Where is negative pi over 2 on a unit circle

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. This circle is called a Unit Circle ; you will get intimately involved with this circle ;.

Again, the coordinate system consists of four quadrants. The main reason we want to work with radians and not degrees is that degrees are sort of artificial numbers and radians have real meaning since they are tied to the circumference of a circle. They are just radians. Notice that when dealing with radians, we have to remember how to get and use Greatest Common Denominators to Add Fractions. Subtracting again will be too small. I have a somewhat simple trick to remember how to do this.

You can see this by converting 90 degrees to radians by using a Unit Multiplier :. Note that you can convert back and forth from degrees to radians in the graphing calculator. So here it is. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile.

The unofficial elections nomination post. Linked 0. Start now and get better math marks! Intro Lesson. Lesson: 1. Intro Learn Practice. What is Unit Circle? Like this blank unit circle below: Blank unit circle with radius of 1. Sine and Cosine evaluated angles of unit circle. Chart of simplified unit circle with sin cos tan sec csc and cot. Do better in math today Get Started Now. Angle in standard position 2. Coterminal angles 3. Reference angle 4. Find the exact value of trigonometric ratios 5.

Unit circle 7. Converting between degrees and radians 8. Trigonometric ratios of angles in radians 9. Radian measure and arc length Law of sines