Trig Identity for cos(t) + sin(t)

This trig identity shows that a combination of sine and cosine functions can be written as a single sine function with a phase shift.

$$a \cos{t} + b \sin{t} = \sqrt{a^2 + b^2} \; \sin(t + \tan^{-1} \frac{a}{b}) $$

for b ≠ 0 and − π2 < tan−1 ab < π2

(Note that tan−1 means arctan .) The phase shift is the quantity tan−1 ab , it has the effect of shifting the graph of the sine function to the left or right.

To derive this trig identity, we presume that the combination a cos(t) + b sin(t) can be written in the form c sin(K + t) for unknown constants c, K .

a cos(t) + b sin(t) = c sin(K + t)
a cos(t) + b sin(t) = c sin(K) cos(t) + c cos(K) sin(t)

We used the formula for sine of a sum of angles to expand the right hand side above. To have equality for any value of t , the coefficients of cos(t) and sin(t) must be equal on the left and right sides of the equation.

a = c sin(K)
b = c cos(K)

Solving this system of simultaneous equations leads us to

c = ± √(a2 + b2)
K = tan−1 ab

So the trig identity for b ≠ 0 is

a cos(t) + b sin(t) = ± √(a2+b2) sin(t + tan−1 ab)

If we limit the arctan to be within

π2 < tan−1 ab < π2

then we can always choose the + in front of the square root.

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