express 20sin theta + 4 cos theta as R sin(theta + alpha)
R sin(theta + alpha) =
R cos(alpha)sin(theta) +
R sin(alpha)cos(theta)
---->
Rcos(alpha) = 20
Rsin(alpha) = 4
The x-y coordinates of a point on a circle of radius R that makes an angle og alpha with the positive x-axis are with radius R are R cos(alpha) and R sin(alpha) respectively.
R follows from:
20^2 + 4^2 = R^2
If you divide the two equations you get tan(alpha)
To express the expression 20sin(theta) + 4cos(theta) as R sin(theta + alpha), we need to find the values of R and alpha.
Let's start by comparing the given expression with the general form R sin(theta + alpha):
20sin(theta) + 4cos(theta) = R sin(theta + alpha)
Now, let's compare the coefficients of sin(theta) and cos(theta) on both sides:
20 = R cos(alpha) (Coefficient of sin(theta))
4 = R sin(alpha) (Coefficient of cos(theta))
From these equations, we can solve for R and alpha.
Rcos(alpha) = 20 (Equation 1)
Rsin(alpha) = 4 (Equation 2)
Now let's consider the x-y coordinates of a point on a circle of radius R that makes an angle alpha with the positive x-axis. The x-coordinate is R cos(alpha), and the y-coordinate is R sin(alpha).
By Equation 1: Rcos(alpha) = 20
This gives us the value of R cos(alpha), which represents the x-coordinate of the point on the circle.
By Equation 2: Rsin(alpha) = 4
This gives us the value of R sin(alpha), which represents the y-coordinate of the point on the circle.
So, R represents the radius of the circle, and (R cos(alpha), R sin(alpha)) gives us the x-y coordinates of a point on the circle.
To find the value of R, we can use the Pythagorean theorem. The radius of the circle can be found by the equation:
R^2 = (R cos(alpha))^2 + (R sin(alpha))^2
R^2 = (20)^2 + (4)^2
R^2 = 400 + 16
R^2 = 416
Taking the square root of both sides, we get:
R = √416
To find the value of alpha, we can divide Equation 2 by Equation 1:
(R sin(alpha))/(R cos(alpha)) = 4/20
tan(alpha) = 1/5
So, alpha can be found by taking the inverse tangent of (1/5):
alpha = tan^(-1)(1/5)
Therefore, we have expressed the given expression 20sin(theta) + 4cos(theta) as Rsin(theta + alpha), where R = √416 and alpha = tan^(-1)(1/5).