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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