BQ#3 – Unit T Concepts 1-3

Tangent: The proportion for tangent is sin(x)/cos(x). In the first quad, sine and cosine are positive and in that ratio, that also means that tangent is positive. In the second quad, sine is positive and cosine is negative and in that ratio, tangent is negative. In the third quad, sine is negative and so is cosine and that means that tangent is negative. In the fourth quad, sine is negative and cosine is positive and that means that tangent is negative. When cosine is equal to zero, that makes the ratio undefined which also means that there will be an asymptote. The asymptotes for tangent are based on sine and cosine, cosine equals to zero on 90* (pi/2) and 270* (3pi/2). 

Cotangent: The ratio for cotangent is cos(x)/sin(x). Based on ASTC, the first quad is positive, the second is negative, the third is positive, and the fourth is negative. For this ratio, when sine is equaled to zero, there is an asymptote. The asymptotes for cotangent exist on 0 and 180 (pi). 

Secant: The ratio for secant is 1/cos(x). Using the ASTC, cosine in the first quad is positive so secant is positive. In the second quad, cosine is negative so secant is negative. In the third quad, cosine is negative so secant is negative. In the fourth quad, cosine is positive so secant is positive. The asymptotes of secant are based on 90* (pi/2) and 270* (3pi/2) because asymptotes exist when the ratio is undefined which is when cosine is equaled to zero.

Cosecant: The ratio is 1/sin(x). Using the ASTC, sine is positive in the first quad so cosecant is positive. In the second quad, sine is positive so cosecant is also positive. In the third quad, sine is negative so cosecant is also negative. In the fourth quad, sine is negative and so is cosecant. The asymptote is based on the sine so when sine is equaled to zero, that means there is an asymptote. The asymptotes for cosecant exist in 0 and 180 (pi).