For what x does sin 2x = cos 2x?

Question

1. What linear function has f(0) = 5 and f(7) = 12?

2. If f(t) = 2t – t², what is (f(t + h) – f(t)) / h?

3. For what x does sin 2x = cos 2x?

Answer 1

sin(2x) = cos(2x)

 

If you want only solutions between 0 and 2π, proceed as follows:  

2x = π/4 + 2kπ or 5π/4 + 2kπ, where k = 0, ±1, ±2, ...

 

So, x = π/8 + kπ, 5π/8 + kπ

 

If k = 0, we get x = π/8 and 5π/8 

 

If k = 1, we get x = 9π/8 and 13π/8

 

Other choices for k give solutions outside of the interval [0, 2π].  

 

There are a total of 4 solutions between 0 and 2π:  

 

       π/8, 5π/8, 9π/8, and 13π/8

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For problem 1, the graph of a linear function is a line.  

f(0) = 5, means that (0,5) is a point on the line and f(7) = 12 means that (7, 12) is also on the line.                                      

 

slope = (12-5)/(7-0) = 1

Equation of line:  y - 5 = 1(x - 0)

 

                               y = x + 5 

 

 

Answer 2

sin2x = cos2x   

 

 2x has to be pi/4  then x=pi/8 

 in general  2x = pi/4 +2kpi ,where k is an integer  also 2x= 5pi/4 +2kpi

so x = pi/8 , 5pi/8  ,9pi/8 ,and 13pi/8

 

Answer 3

1.   If f(t) = 2t – t², what is (f(t + h) – f(t)) / h?

 

      (f(t + h) – f(t)) / h = (2(t+h)-(t+h)²-2t+t²)/h

          

                                 = (2t+2h-t²-2th-h²-2t+t²)/h

 

                                 =(2h-2th-h²)/h

 

                                 = 2-2t-h

 

As h approaches 0 (f(t + h) – f(t)) / h=2-2t

 

2.     For what x does sin 2x = cos 2x? 

 

        sin θ = cos θ when θ=Π/4 and θ=5Π/4

 

        So 2x = Π/4 and 5Π/4 and x = Π/8 and 5Π/8