college trigonometry

Question

let a be an acute angle in degrees. adding together a's complement, a's supplement and one of a's coterminal angles gives 2019 degrees. determine a.

Answer 1

the complement of a: 90 - a degrees

the supplement of a: 180 - a degrees

a coterminal angle of a: n*360 + a degrees [multiple of 360 degrees larger than angle a]

TOTAL --------------

2019 degrees

(90 - a) + (180 - a) + (n*360 + a) = 2019

630 + n*360 - a = 2019

n*360 - a = 1389

a = n*360 - 1389

Let n = 4 [We need n*360 to be just a little bit larger than 1389]

a = 1440 - 1389

a = 51 degrees [acute is between 0 and 90]

Answer 2

The sum of a's complement and supplement is (90 - a) + (180 - a) = 270 - 2a

There are infinitely many coterminal angles for a to choose from: a + 360, a + 720, a + 1080, etc

If we use the coterminal angle a + 5(360) = a + 1800, then we have:

(270 - 2a) + (a + 1800) = -a + 2070

If -a + 2070 = 2019, then -a = -51. So, a = 51° in that case.

Answer 3

There seems to be a problem as written. Are you sure that the sum of the three angles should be 2019 degrees? This seems very high.

if a is an acute angle, then its measure is between 0 and 90

the complement of a = 90 - a degrees

the supplement of a = 180 - a degrees

a coterminal angle of a = 360 - a degrees OR = 360 + a degrees

Summing these three (two variations) gives:

(90 - a) + (180 - a) + (360 - a) = 630 - 3a

(90 - a) + (180 - a) + (360 + a) = 630 - a

In either case the maximum value for the sum less than 630 degrees, much less than 2019 degrees.