![]() Give yourself plenty of room on the y-axis as the tangent value rises quickly as it nears 90 degrees and jumps to large negative numbers just on the other side of 90 degrees. It may not be fun, but it will help lock it in your mind. Do yourself a favor and plot it out manually at least once using points at every 10 degrees for 360 degrees. This pattern repeats itself every 180 degrees. This portion looks a little like the left half of an upside down parabola. At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. You are left with something that looks a little like the right half of an upright parabola. At 45 degrees the value is 1 and as the angle nears 90 degrees the tangent gets astronomically large. At the angle of 0 degrees the value of the tangent is 0. ![]() When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. The angle line, COT line, and CSC line also forms a similar triangle. A bunch of those almost impossible to remember identities become easier to remember when the TAN and SEC become legs of a triangle and not just some ratio of other functions. You can also see that 1/COS = SEC/1 and 1^2 + TAN^2 = SEC^2. When you compare the sine leg over the cosine leg of the first triangle with the similar sides of the other triangle, you will find that is equal to the tangent leg over the angle leg. In this second triangle the tangent leg is similar to the sin leg the angle leg is similar to the cosine leg and the secant leg (the hypotenuse of this triangle) is similar to the angle leg of the first triangle. The second bonus – the right triangle within the unit circle formed by the cosine leg, sine leg, and angle leg (value of 1) is similar to a second triangle formed by the angle leg (value of 1), the tangent leg, and the secant leg. You can, with a little practice, “see” what happens to the tangent, cotangent, secant and cosecant values as the angle changes. As the angle nears 90 degrees the tangent line becomes nearly horizontal and the distance from the tangent point to the x-axis becomes remarkably long. When the angle is close to zero the tangent line is near vertical and the distance from the tangent point to the x-axis is very short. Some people can visualize what happens to the tangent as the angle increases in value. The sign of that value equals the direction positive or negative along the y-axis you need to travel from the origin to that y-axis intercept. The distance from the origin to where that tangent line intercepts the y-axis is the cosecant (CSC). The sign of that value equals the direction, positive or negative, along the x-axis you need to travel from the origin to that x-axis intercept. You will find that the TAN and COT are positive in the first and third quadrants and negative in the second and fourth quadrants.Īs a bonus, the distance from the origin (point (0,0)) to where that tangent line intercepts the x-axis is the secant (SEC). ![]() For example, If the line intersects the negative side of the x-axis and the positive side of the y-axis, you would multiply the length of the tangent line by (-1) for the x-axis and (+1) for the y-axis. To determine the sign (+ or -) of the tangent and cotangent, multiply the length of the tangent by the signs of the x and y axis intercepts of that “tangent” line you drew. If you extend the tangent line to the y-axis, the distance of the line segment from the tangent point to the y-axis is the cotangent (COT). The distance of this line segment from its tangent point on the unit circle to the x-axis is the tangent (TAN). This line is at right angles to the hypotenuse at the unit circle and touches the unit circle only at that point (the tangent point). Using the unit circle diagram, draw a line “tangent” to the unit circle where the hypotenuse contacts the unit circle. I do not understand why Sal does not cover this. While you are there you can also show the secant, cotangent and cosecant. I think the unit circle is a great way to show the tangent.
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