what is the maximum value of sinthetacostheta?
let y = sinØ cosØ
dy/dØ = sinØ(-sinØ) + cosØ(cosØ)
= cos^2 Ø - sin^2 Ø
= cos 2Ø
but for a max/min, dy/dØ = 0
cos 2Ø = 0
2Ø = π/2 or 3π/2 , (90° or 270°)
Ø = π/4 or 3π/4 , (45° or 135°)
when Ø = π/4
sin(π/4) (cos(π/4) = (1/√2)(1/√2) = 1/2
for Ø = 3π/2 we get -1/2
so the max is 1/2
seems like a lot of work
y = 1/2 sin(2θ)
max value is thus 1/2
Yup, you got me!
one big DUH!
To find the maximum value of the expression sin(theta)cos(theta), we can utilize the properties of trigonometric functions and basic calculus.
Step 1: We can rewrite the expression as sin(2theta)/2. This is done using the double-angle formula for sine.
Step 2: Since the sine function oscillates between -1 and 1, we can conclude that the maximum value of sin(2theta) is 1.
Step 3: After substituting sin(2theta) = 1 into the expression sin(2theta)/2, we get 1/2.
Hence, the maximum value of sin(theta)cos(theta) is 1/2.