4.4 Trigonometric Functions of Any Angle

Earlier we learned that if θ  is an acute angle there are a set of definitions of trigonometric functions. Now we are going to learn the extension of these definitions so that θ  can be any angle.

Let θ  be an angle in standard position with (x, y) a point on the terminal side of θ  and   

sin θ = y/r
cos θ = x/r                                           
tan θ = y/x, x≠0
                
csc θ = r/y, y≠0
sec θ = r/x, x≠0
cot θ = x/y, y≠0

As shown above, each trigonometric function has its own reciprocal. They come in pairings, which are as follows: sin & csc, cos & sec, and tan & cot.

The signs of these functions can be determined from the definitions of the functions as well.

• IE: Because cos θ  = x/r, cos θ will be positive as long as x is positive.
• Doing this with all trig functions will result in:

Reference angles are the acute angles that correspond with angles greater than 90°. They are formed by the horizontal axis and whatever the terminal side of θ is.

-If θ  is in quadrant I, it is an acute angle so reference so angle θ  has the same measure as reference angle θ'  .

-If θ  is in quadrant II, you would use these equations to find the reference angle θ' :

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-If θ  is in quadrant III, you would use these equations to find the reference angle θ' :

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-If θ  is in quadrant IV, you would use these equations to find the reference angle θ' :

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-Reference angles may have varying signs from their original angles but the sign of the angle can be determined by which quadrant the angle is found in.