These three ratios are the most common trig ratios. The trig (short for trigonometry) ratios sine(sin), cosine (cos) and tangent (tan) are based on properties of right triangles. The unit circle with critical values labeled in radian and degrees angles is shown below: In addition all whole number multiples will also be critical values for all angles ≤ 360°. If you remember that 180° = π radians, it is easy to remember the other angles. The critical values in Quadrant I are as follows: These angle measures are equivalent to radian measures of and, respectively. The word trigonometry comes from the Greek words for “triangle measure Angles whose measures are multiples of 30° and 45° are commonly used in trigonometry. Trigonometry is the study of angles and triangles.
S top!Go to Questions #10-17 about this section, then return to continue on to the next section.
You can convert from degrees to radians and visa versa.Įxample #1: Convert 40° from degrees to radians and radians to degrees. Thus, an angle of 180° = π radians and 90° =. Therefore an angle representing one complete revolution of the circle measures 2π radians or 360°. So the circumference of a unit circle is 2π(1) or 2π radians. The circumference of any circle is 2π r, where r is the radius of the circle. The radian measure of an angle is equal to the length of the arc on the unit circle (a circle centered at the origin with a radius of 1) that is intercepted by the angle in standard position. Another unit of angle measure is called the radian, which is better suited for certain mathematical development, scientific work and engineering applications. Plug and chug.S top!Go to Questions #6-9 about this section, then return to continue on to the next section.ĭegree measure of angles is used extensively in engineering, surveying and navigation. Much better: 120° falls into Quadrant II. Since 480° is larger than 360°, we've gotta subtract 360° to get our ɵ.
We've got an angle that's between 270° and a full 360°, so we're in Quadrant IV this time. In Quadrant II, our formula is ρ = 180° – ɵ. We wanna turn that into an acute angle that's closer to the negative x-axis, because that angle will be smaller and easier to deal with. That's 135° measured from the positive x-axis. Since 135° is more than 90° but less than 180°, our angle is in Quadrant II. Sample Problemįirst off, we need to figure out which quadrant we're in. Now, let's see what we can do with our newfound knowledge. And no, you don't have to take calculus-yet. In Quadrant IV, only cosine and its reciprocal function secant are +.Īn easy way to remember this is ASTC (All, Sine, Tangent, Cosine), or All Students Take Calculus.Ĭalm down it's just a mnemonic device.
In Quadrant III, only tangent and its reciprocal function cotangent are +. The other four trig functions are negative. In Quadrant II, only sine and its reciprocal function cosecant are +. In Quadrant I, all six trig functions are positive. Now let's look at our six trig functions and see what their signs are for each quadrant. If ɵ doesn't fall into this range, then we must add or subtract 360° or 2π, until we have a ɵ in the correct range. One little thing, though: our ɵ has some special requirements. In other words, to turn ɵ into ρ, we subtract ɵ from 180° (or from π radians if we're in radian mode). Our reference angle is ρ = ɵ, because it's already an acute angle. No need for anything fancy in Quadrant I. Since we're on a Greek fix, we'll use ɵ ("theta") to represent the actual angle. Let's use a ρ to represent our reference angles, which is just the common Greek letter "rho." As in, "Rho, rho, rho your boat." It's the smallest angle that our angle makes with the x-axis. Reference AnglesĪ reference angle is just the acute version of whatever angle we're looking at. To do this, we first need to learn all about reference angles. To simplify trigonometric expressions, we often rewrite non-acute angles as acute angles. Angles larger than 90° fall into one of the other three quadrants. We can also see that 180° sits right between Quadrant II and Quadrant III, and 270° separates Quadrant III and Quadrant IV.Īcute angles (that is, smaller than 90°…and adorable) fall into Quadrant I. That's because 90° is exactly one-quarter of a full circle. Notice how 90° is right there at the positive y-axis. The coordinate plane is split into four sections or quadrants, like so.