write the expression as a function of an acute angle whose measure is less than 45.
a. sin 80
b. sin (-100)
To express the given expressions as functions of an acute angle whose measure is less than 45 degrees, we can make use of two fundamental trigonometric properties: the periodicity of sine and cosine functions and the reference angle concept.
a. sin 80:
We know that the sine function is periodic with a period of 360 degrees (or 2π radians). Therefore, we can express sin 80 as sin (80 - 360 degrees) or sin (80 - 2π radians) or sin (-280 degrees) or sin (-280 - 2π radians) since these angles are equivalent to 80 degrees.
To find an acute angle less than 45 that is equivalent to 80 degrees, we can deduct 360 degrees repeatedly until we reach a value within the desired range. Here's the step-by-step process:
80 degrees - 360 degrees = -280 degrees
-280 degrees - 360 degrees = -640 degrees
Since -640 degrees is still not within the desired range, we continue this process:
-640 degrees - 360 degrees = -1000 degrees
-1000 degrees - 360 degrees = -1360 degrees
Finally, we find an angle less than 45 degrees:
-1360 degrees - 360 degrees = -1720 degrees
Therefore, sin 80 can be expressed as sin (-1720 degrees) or sin (-1720 - 2π radians).
b. sin (-100):
Similarly, we can use the periodicity of the sine function to express sin (-100) as sin (-100 + 360 degrees) or sin (-100 + 2π radians) since these angles are equivalent to -100 degrees.
To find an acute angle less than 45 that is equivalent to -100 degrees, we can add 360 degrees repeatedly until we reach a value within the desired range. Here's the step-by-step process:
-100 degrees + 360 degrees = 260 degrees
260 degrees + 360 degrees = 620 degrees
Since 620 degrees is still not within the desired range, we continue this process:
620 degrees + 360 degrees = 980 degrees
980 degrees + 360 degrees = 1340 degrees
Finally, we find an angle less than 45 degrees:
1340 degrees + 360 degrees = 1700 degrees
Therefore, sin (-100) can be expressed as sin (1700 degrees) or sin (1700 + 2π radians).
Note: The expressions obtained in each case still represent the same value as the original expression but are expressed using an acute angle less than 45 degrees.