If you’re not sure how to figure this out, draw a diagram to help you determine whether the length of the line will be short, medium, or long. Say, for example, you get a problem that looks like this:Īgain, using the chart above, we can see that the x-coordinate (or cosine) for $/2$. By looking at this chart, we can see that the y-coordinate is equal to $1/2$ at 30°. And since the y-coordinate equals sine, our answer is as follows:īut what if you get a problem that uses radians instead of degrees? The process for solving it is still the same. Where do you start? Let’s take a look at the unit circle chart again-this time with all major angles (in both degrees and radians) and their corresponding coordinates:ĭon’t get overwhelmed! Remember, all you’re solving for is $\sin30°$. This slogan definitely applies if you're not a math lover.Īs stated above, the unit circle is helpful because it allows us to easily solve for the sine, cosine, or tangent of any degree or radian. It's especially useful to know the unit circle chart if you need to solve for certain trig values for math homework or if you're preparing to study calculus.īut how exactly can knowing the unit circle help you? Let's say you’re given the following problem on a math test-and are not allowed to use a calculator to solve it: This gives us the following values for sine and cosine: Here, we can see that the x-coordinate equals 0 and the y-coordinate equals 1. What if the angle is 90° and makes a perfectly vertical line along the y-axis? We know that the cosine is equal to the x-coordinate, and the sine is equal to the y-coordinate, so we can write this: On this line, the x-coordinate equals 1 and the y-coordinate equals 0. Here is an overview of all major angles in degrees and radians on the unit circle:īut what if there’s no triangle formed? Let’s look at what happens when the angle is 0°, creating a horizontal straight line along the x-axis: Since $1^2=1$, we can simplify this equation like this:īe aware that these values can be negative depending on the angle formed and what quadrant the x- and y-coordinates fall in (I’ll explain this in more detail later). Therefore, we can say that the formula for any right triangle in the unit circle is as follows: We know that the cosine of an angle is equal to the length of the horizontal line, the sine is equal to the length of the vertical line, and the hypotenuse is equal to 1. Why is all of this important? Remember that you can solve for the lengths of the sides of a triangle using the Pythagorean theorem, or $a^2 b^2=c^2$ (in which a and b are the lengths of the sides of the triangle, and c is the length of the hypotenuse). ( The triangle’s longest line, or hypotenuse, is the radius and therefore equals 1.) In other words, cosine = x-coordinate, and sine = y-coordinate. On this triangle, the cosine is the horizontal line, and the sine is the vertical line. The unit circle, or trig circle as it’s also known, is useful to know because it lets us easily calculate the cosine, sine, and tangent of any angle between 0° and 360° (or 0 and 2π radians).Īs you can see in the above diagram, by drawing a radius at any angle (marked by ∝ in the image), you will be creating a right triangle. Typically, the center point of the unit circle is where the x-axis and y-axis intersect, or at the coordinates (0, 0): (This also means that the diameter of the circle will equal 2, since the diameter is equal to twice the length of the radius.) This means that for any straight line drawn from the center point of the circle to any point along the edge of the circle, the length of that line will always equal 1. The unit circle is a circle with a radius of 1. We also give you three tips to help you remember how to use the unit circle. In this article, we explain what the unit circle is and why you should know it. But how does it work? And what information do you need to know in order to use it? The unit circle is an essential tool used to solve for the sine, cosine, and tangent of an angle. If you’re studying trig or calculus-or getting ready to-you’ll need to get familiar with the unit circle.
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