![]() When the terminal side is in the third quadrant (angles from 180° to 270°), our reference angle is our given angle minus 180°. So, if our given angle is 110°, then its reference angle is 180° – 110° = 70°. When the terminal side is in the second quadrant (angles from 90° to 180°), our reference angle is 180° minus our given angle. So, if our given angle is 33°, then its reference angle is also 33°. This makes sense, since all the angles in the first quadrant are less than 90°. When the terminal side is in the first quadrant (angles from 0° to 90°), our reference angle is the same as our given angle. How we find the reference angle depends on the quadrant of the terminal side. How do We Find the Reference Angle without a Calculator? As for the sign, remember that Sine is positive in the 1st and 2nd quadrant and Cosine is positive in the 1st and 4th quadrant. Once we know their sine, cosine, and tangent values, we also know the values for any angle whose reference angle is also 45° or 60°. This is useful for common angles like 45° and 60° that we will encounter over and over again. The sign may not be the same, but the value always will be. ![]() The reference angle always has the same trig function values as the original angle. Notice how the second ray is always on the x-axis. Here’s an animation that shows a reference angle for four different angles, each of which is in a different quadrant. It’s always the smaller of the two angles, will always be less than or equal to 90°, and it will always be positive. This second angle is the reference angle. If we draw it to the left, we’ll have drawn an angle that measures 36°. If we draw it from the origin to the right side, we’ll have drawn an angle that measures 144°. Our second ray needs to be on the x-axis. But we need to draw one more ray to make an angle. Now we have a ray that we call the terminal side. We draw a ray from the origin, which is the center of the plane, to that point. We keep going past the 90° point (the top part of the y-axis) until we get to 144°. We rotate counterclockwise, which starts by moving up. We start on the right side of the x-axis, where three o’clock is on a clock. Let’s say we want to draw an angle that’s 144° on our plane. Go back to Calculators page What is a Reference Angle, anyway?
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