Question 1208134: Q:1- What is the maximum and minimum value of Sin(x) + Cos(x)?
Q;2- What is the maximum and minimum value of Sin(x) - Cos(x)?
Q:3- What is the maximum and minimum value of Tan(x)/Sin(x) in [π/4 to π/3]?
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
I'll do question 1 to get you started.
In the future please only post one question at a time.
If you are familiar with calculus, then,
f(x) = sin(x) + cos(x)
f ' (x) = cos(x) - sin(x) ..... apply derivative
The local max occurs when f ' (x) = 0.
cos(x) - sin(x) = 0
cos(x) = sin(x)
Use the unit circle to determine that the sine and cosine values are equal when x = pi/4 when in quadrant I.
Use this x value to determine the max value of f(x).
f(x) = sin(x) + cos(x)
f(pi/4) = sin(pi/4) + cos(pi/4)
f(pi/4) = sqrt(2)/2 + sqrt(2)/2
f(pi/4) = sqrt(2)
sqrt(2) is the max value of sin(x)+cos(x).
Through symmetry, the min value of that expression is -sqrt(2)
sqrt(2) = 1.41421 approximately
If you are not familiar with calculus, then you'll need to rely solely on a graphing calculator.
The graphing calculator should have functions called "minimum" and "maximum".
Here is an article that talks about finding the local min and max on a TI84
https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Tradler_and_Carley)/04%3A_Introduction_to_the_TI-84/4.02%3A_Finding_zeros_maxima_and_minima
Another approach is shown here
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1046265.html
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