Trigonometric Ratios and Identities

To find the maximum value of the expression 12sin‌x−5‌cos‌x+3, we start by considering the form asin‌x+b‌cos‌x, which achieves its maximum value at √a2+b2.
For our specific expression, we have a=12 and b=−5.
The calculation for the maximum value of 12sin‌x−5‌cos‌x is:
√122+(−5)2=√144+25=√169=13
Therefore, the maximum value of the entire expression 12sin‌x−5‌cos‌x+3 is:
13+3=16