![]() ![]() How far above the ground is the mark at this point? A small mark is painted at the very top of the tire, and then the tire is rolled forward slightly so that the mark rotates through an angle of pi/4 radians. Practice Problem: A tire has a radius (outer radius) of 10 inches. Note that to compute trig functions using radians instead of degrees, you must make sure your calculator is in radian mode, not degree mode. Below are the sine, cosine, and tangent functions plotted for the θ domain. As the angle θ changes, these distances change we can plot them graphically, however, to see how they behave. The tangent of the angle, tan θ, is simply the ratio of these distances (specifically, ). Sin θ = distance of point P from the x-axisĬos θ = distance of point P from the y-axis What we've also done is define two functions: In other words, point Phas coordinates (cos θ, sin θ), where θ is the angle formed between the x-axis and the radius to P. Note the following, where we apply what we learned about right-triangle trig: Let's look again at our unit circle.Įach point P on the unit circle, designated by the coordinates ( x 1, y 1), can be expressed in terms of the angle θ as well. Now, we can finally look at what circles have to do with trigonometry. Again, only the calculation for part a is shown below the rest are similar. Solution: Here, simply use the same process as the previous practice problem, but use the reciprocal ratio of radians and degrees. ![]() Practice Problem: Convert each of the following angle measures from radians to degrees. Note that the negative sign (part c) translates directly from degrees to radians.Ī. ![]() Below is the calculation for part a the other parts follow the same pattern. We can use this ratio to convert from degrees to radians. Solution: First, note that 360° is equal to 2 π radians. At those spots, the function is undefined, like the sound of one hand clapping.Interested in learning more? Why not take an online Precalculus course? Tangent, for instance, would divide by zero whenever cosine equals zero, at 0 and π radians. All four trig functions other than sine or cosine involve division, so there is the risk of the ever-dreaded division by zero. There is a danger to finding the angles of trig functions. We don't need a whole new sign language to work with them. They tend to balance each other out.Ĭosecant and secant have the same sign as sine and cosine, respectively. Quadrant V exists outside of our dimension, and constantly produces ice cream and volcanoes. In order for cotangent (or tangent) to be positive, both cosine and sine must be either positive or negative together, like in Quadrants I and III. If trig functions were magnets, we'd have a hard time pulling them apart now. We're dividing cosine by sine, so in the second quadrant we're dividing a negative (cosine) by a positive (sine). That means the value of cot is, but now we need its sign. We stare at our left hand for a little bit, and discover that cos is and sin is. Looking back at that list we made just a minute ago, we see that: The angle is a little less than π, so it's in the second quadrant. How about we solve a problem? We're thinking…. The key to finding angles for these functions is to know that they're all knock-off brand soda to sine and cosine's Coke and Pepsi. If they worked different ways, that'd be four times the work, and we'd honestly flake out before finishing. We've also got cosecant, secant, and cotangent to do as well. Like a superhero, but Shmoopier.Īnyway, that's pretty tangential to what we're here to talk about, which is the tangent of different angles. We'd fly around fighting crime, and bears, and criminal bears. You know what would make us really happy? Jetpacks. ![]()
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