Up to now in this Precalculus course, only two trigonometric
functions have been studied extensively—sine and cosine.
There's a reason for this!
All the remaining trigonometric functions are defined in terms of sine and cosine.
This section discusses the four remaining trigonometric functions: tangent, cotangent, secant, and cosecant.
DEFINITIONS
tangent, cotangent, secant, cosecant
Let $\,t\,$ be a real number (restrictions are noted individually below).
Think of $\,t\,$, if desired, as the radian measure of an angle. Then:

Cosecant is the reciprocal of the sine. Secant is the reciprocal of the cosine. Cotangent is the reciprocal of the tangent. How might you remember these relationships? The memory device at right should help! Write the trig functions vertically, with function followed immediately by cofunction: Then, connect the functions as shown—they're reciprocals! 

Consider an acute angle $\,\theta\,$ laid off in the unit circle.
Its terminal point is shown in black below.
Only one auxiliary triangle (in yellow below—it overlaps the green triangle)
is needed to explain
the significance of the names ‘tangent’ and ‘secant’, as follows:
The green and (overlapping) yellow triangles below are similar. Why? They share the angle $\,\theta\,$ and they both have a right angle. 

The word ‘tangent’ derives from the Latin tangens, meaning ‘touching’. The tangent gives the length of a segment that is tangent to the unit circle.


green triangle:  $\cos\theta$  $\sin\theta$  
yellow triangle:  $1$  $y$ (initially unknown) 
bottom of triangle  hypotenuse  
green triangle:  $\cos\theta$  $1$ 
yellow triangle:  $1$  $h$ (initially unknown) 
In English, to complement (not compliment!) is to fill out or to complete.
By definition, the complement of an angle $\,\theta\,$ (given in degree measure) is $\,90^\circ  \theta\,$.
Similarly, the complement of an angle $\,\theta\,$ (given in radian measure) is $\,\frac{\pi}2  \theta\,$.
Thus, an (acute) angle and its complement, together, fill out a right angle.
The ‘co’ in cosine, cotangent, and cosecant refers to ‘complement’.
The following relationships (when they're defined) are true for all real numbers $\,\theta\,$ (not just those in the first quadrant):
By definition: $$\cos\theta = \ell$$ Using the right triangle definition: $$\displaystyle\sin(\frac{\pi}2  \theta) = \frac{\text{OPP}}{\text{HYP}} = \frac{\ell}{1} = \ell$$ Comparing: $$\cos\theta = \sin(\frac{\pi}{2}  \theta)$$ 
By definition: $$ \begin{gather} \cos\theta = \ell\cr \sin\theta = m\cr \cot\theta := \frac{\cos\theta}{\sin\theta} = \frac{\ell}{m} \end{gather} $$ Using the right triangle definition: $$\displaystyle\tan(\frac{\pi}2  \theta) = \frac{\text{OPP}}{\text{ADJ}} = \frac{\ell}{m}$$ Comparing: $$\cot\theta = \tan(\frac{\pi}{2}  \theta)$$ 
By definition: $$ \begin{gather} \sin\theta = m\cr \csc\theta := \frac{1}{\sin\theta} = \frac{1}{m} \end{gather} $$ Using the definition of secant and the right triangle definition of cosine: $$\displaystyle\sec(\frac{\pi}2  \theta) := \frac{1}{\cos(\frac{\pi}2  \theta)} = \frac{1}{m/1} = \frac{1}{m}$$ Comparing: $$\csc\theta = \sec(\frac{\pi}{2}  \theta)$$ 
Here are two excellent references for historical and cultural information on the trigonometric functions:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
