SP#7: Unit Q Concept 2: Finding All Trig Functions When Given OneTrigFunction and Quadrant

This SP7 was made in collaboration Anahi C. Please visit the other awesome posts on their blog by going here.

We are given two trig functions to help solve the equation: cos(x) = 8rad89/89 and cot(x) = 8/5. 

In the photo above, we see the use of the reciprocal identities to find tan and sec. Thanks to our identities, we know that tan(x) = 1/cot(x). Since we are given cot, all we have to do is plug in the information to the formula and we receive the trig function of tan. The same process goes towards finding sec as sec(x) = 1/cos(x). Again we plug in the information that has been supplied to come by our answer. We then used the Pythagorean identity of "1-csc^2(x) = -cot^2" to find the value of csc. The negatives cancel out and again we fill in the formula with the information we know. Knowing when to do what (square, divide, etc.) is key to correctly solving this type of identity.

In order to find sin, we once again use the reciprocal identity that equates "sin(x) = 1/cot(x)." Once again we simply fill this out with the information we received from earlier and given steps. It is important that we multiply by the reciprocal in order to receive the correct answer! 

This step shows the use of the inverse trig functions from Unit O. Csc, sec, and cot are simply the reciprocals of the sinc, cos, and tan. Since we already know much of our necessary information, all we have to do is fill in the formulas! csc = r/y -- sec = r/x -- cot = x/y

These steps show the use of the regular trig functions that we learned in Unit O. We fill in the formulas with what we know, simplify, and rationalize. It is important that there are NO square roots in your denominators! sin = y/r -- cos = x/r -- tan = y/x

Here we have our final triangle. In the upper right hand corner, we can see which quadrant the triangle lies within. Since we were given cos and cot, both of which were positive, we were given information that can eliminate possible quadrants. cos must stay within the I and III quadrant while cot must do the same. With all of the info we've found, we can see the final result of the triangle.