How do you sketch a right triangle corresponding to cottheta=5 and find the third side, then find the other five trigonometric functions?

1 Answer
A. S. Adikesavan
Jan 14, 2017

tan theta =1/5, (sin theta, cos theta)=+-1/sqrt26(1, 5) and, correspondingly, (csctheta, sectheta)=+-sqrt26(1, 1/5). The right angled triangle OAB keeps off negative sign.

Explanation:

#cot theta = 5 to the ratio (adjacent side)/(opposite side) = 5

Taking the opposite side length as 1 unit,

the third side = sqrt(1^2+5^2)=sqrt 26 units#.

Now, keep off the notion of associating theta with a right angled

triangle.

As cot theta > 0, theta in Q_1 or Q_3.

tan theta = 1/cot theta=1/5

As sine and cosine are both positive in Q_1 and both negative in

Q_3, sin theta=1/5sqrt(1-sin^2theta), giving sin^2theta=1/26, and so,

sin theta = +-1/sqrt 26,

Likewise, with the same sign,

cos theta = +-5/sqrt 26

Combining these two results,

(sin theta, cos theta)=+-1/sqrt26 (1, 6)

Correspondingly, the reciprocals

(csc theta, sec theta) = +-sqrt 26(1. 1/5)

The triangle OAB asked for has vertices O(0, 0), A(5, 0) and B(0,

1), with angleOAB = cot^(-1)5 and the third side AB = sqrt 26. I

leave making graph to the reader

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