## the points of discontinuty of a function f(x)=[x], xe(0, 3)​

Question

the points of discontinuty of a function f(x)=[x], xe(0, 3)​

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7 months 2021-10-07T21:42:59+00:00 1 Answer 0 views 0

1. Answer: Sine and cosine have range ±1. They never hit their extrema at the same time so the sum is never ±2. So really we have three different equations, cosx+sinx=−1, cosx+sinx=0, cosx+sinx=1. f(x) will have the value −2 in between a couple of places where cosx+sinx=−1.

The linear combination of a sine and cosine of the same angle is a rotation and a scaling. When there’s the same coefficient on cosine and sine (here 1), that’s a rotation of 45∘. We note

cos45=sin45=1/2–√

2–√cos45=2–√sin45=1

cosx+sinx=k

2–√cos45cosx+2–√sin45sinx=k

2–√cos(45−x)=k

cos(45−x)=k/2–√

k=0 first

cos(45−x)=0=cos(90)

45−x=±90−360k

x=45±90+360k

From 0 to 360, that’s x=135 and x=360−45=315

k=1

cos(45−x)=1/2–√=cos(45)

45−x=±45−360k

x=45±45+360k

x=0 or x=90

k=−1

cos(45−x)=−1/2–√=cos(135∘)

45−x=±135−360k

x=45±135+360k

x=270 or x=180

So we found angle:exact-value for cosx+sinx of

0:190:1135:0180:−1270:−1315:0

Step-by-step explanation: