Problem 1 Trigonometric Values Beyond the Unit Circle

For a standard-position angle determined by the point ( 8 , − 6 ) , open 8 comma negative 6 close comma what are the values of the six trigonometric functions?

First, find the distance r of the point from the origin:

r = ( 8 − 0 ) 2 + ( − 6 − 0 ) 2 = 10 r equals . square root of open , 8 minus 0 , close squared . plus . open , negative 6 minus 0 , close squared end root . equals 10

Then,

sin θ = y r = − 6 10 = − 3 5 cos θ = x r = 8 10 = 4 5 tan θ = y x = − 6 8 = − 3 4 csc θ = r y = 10 − 6 = − 5 3 sec θ = r x = 10 8 = 5 4 cot θ = x y = 8 − 6 = − 4 3 table with 6 rows and 1 column , row1 column 1 , sine theta equals , y over r , equals , negative 6 over 10 , equals negative , 3 fifths , row2 column 1 , cosine theta equals , x over r , equals , 8 tenths , equals , 4 fifths , row3 column 1 , tangent theta equals , y over x , equals , negative 6 over 8 , equals negative , 3 fourths , row4 column 1 , co-secant theta equals , r over y , equals , 10 over negative 6 , equals negative , 5 thirds , row5 column 1 , secant theta equals , r over x , equals , 10 over 8 , equals , 5 fourths , row6 column 1 , co-tangent theta equals , x over y , equals , 8 over negative 6 , equals negative , 4 thirds , end table

✓ Got It?

1. For a standard-position angle determined by the point ( − 2 , 12 ) , open negative 2 comma 12 close comma what are the values of the six trigonometric functions?

Take Note Key Concept: Trigonometric Ratios for a Right Triangle

If θ theta is an acute angle of a right triangle, x is the length of the adjacent leg (ADJ), y is the length of the opposite leg (OPP), and r is the length of the hypotenuse (HYP), then the trigonometric ratios of θ theta are as follows.

sin θ = y r = OPP HYP cos θ = x r = ADJ HYP tan θ = y x = OPP ADJ csc θ = r y = HYP OPP sec θ = r x = HYP ADJ cot θ = x y = ADJ OPP table with 6 rows and 1 column , row1 column 1 , sine theta equals , y over r , equals , fraction

Image Long Description