write y=sinx-cosx in the form y=ksin (x+a), where the measure of a is in radians.
The coefficients of sin(x) and -cos(x) are 1 and -1 respectively.
We divide each term by the factor sqrt(1²+(-1)²)=sqrt(2)
y=sinx-cosx
=>
y=sqrt(2)[sin(x)*(1/sqrt(2)+cos(x)*(-1/sqrt(2))
Since
cos(-π/4)=1/sqrt(2), and
sin(-π/4)=-1/sqrt(2),
we set a=3π/2, then we can rewrite the above as:
y=sqrt(2)*[sin(x)cos(a)+cos(x)sin(a)]
=sqrt(2)sin(x+a)
where k=sqrt(2) and a=-π/4
To rewrite the equation y = sin(x) - cos(x) in the form y = ksin(x + a), we need to find values for k and a.
Let's start by applying the trigonometric identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b).
We can rewrite the equation as follows:
y = √2 * (sin(x)cos(π/4) - cos(x)sin(π/4))
Now, using the values from the identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b), we can simplify the equation further:
y = √2 * sin(x - π/4)
Therefore, we have rewritten y = sin(x) - cos(x) in the form y = ksin(x + a), where k = √2 and a = -π/4.
To write the equation y = sinx - cosx in the form y = ksin(x + a), we'll first start by rearranging the equation.
Let's express sinx - cosx as a single trigonometric function using the relationship between sine and cosine. We know that sin(a + b) = sin(a)cos(b) + cos(a)sin(b).
Applying this formula, we can rewrite sinx - cosx as sin(x + π/4 - π/2), where π/4 is the value of a.
Now, let's simplify the expression further:
sin(x + π/4 - π/2)
Using the formula for the sine of the difference of two angles, sin(a - b) = sin(a)cos(b) - cos(a)sin(b), we can rewrite the expression as:
sin(x + π/4)cos(π/2) - cos(x + π/4)sin(π/2)
Since cos(π/2) = 0 and sin(π/2) = 1, the equation becomes:
sin(x + π/4) - cos(x + π/4)
Finally, we can rewrite it in the desired form:
y = sin(x + π/4)
Thus, y = ksin(x + a) with a = π/4.