Write cos (lambda + pi/3) as a function of lambda only.
To write cos (lambda + pi/3) as a function of lambda only, we can use the trigonometric identity known as the sum of angles formula. This formula states that cos(x + y) = cos(x) * cos(y) - sin(x) * sin(y).
In this case, if we let x = lambda and y = pi/3, we can rewrite the expression as follows:
cos(lambda + pi/3) = cos(lambda) * cos(pi/3) - sin(lambda) * sin(pi/3)
Now, we need to express cos(pi/3) and sin(pi/3) in terms of lambda only.
To do this, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in a coordinate plane. In trigonometry, we use the unit circle to define the values of cosine and sine for different angles.
For pi/3, we can see on the unit circle that the x-coordinate is 1/2 and the y-coordinate is sqrt(3)/2. Therefore, we have:
cos(pi/3) = 1/2
sin(pi/3) = sqrt(3)/2
Substituting these values back into the original expression, we get:
cos(lambda + pi/3) = cos(lambda) * (1/2) - sin(lambda) * (sqrt(3)/2)
So, cos (lambda + pi/3) as a function of lambda only becomes:
cos (lambda + pi/3) = cos(lambda)/2 - (sqrt(3) * sin(lambda))/2