Linear Combination Property

A linear combination of a cosine and sine with equal periods is a single cosine function with the same period but with a phase displacement and a different amplitude. The sketch on this page will help you compute the amplitude and phase displacement of the linear combination of two sinusoidal functions.

Sketch

The sketch below shows graphs of the sinusoidal functions c cos θ, d sin θ, and c cos θ + d sin θ. There is a point P on the graph of c cos θ + d sin θ with its coordinates given. You can move point P and its coordinates will update.

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Investigate

1. Drag sliders c and d, and observe the red graph of the function c cos θ + d sin θ. What type of graph does it have?
2. How can you use point P to determine the amplitude of the function c cos θ + d sin θ? How can you use point P to determine the phase displacement of the function c cos θ + d sin θ, as it relates to the untranslated cosine function?
3. Use the sliders to set c = 3 and d = 4. Drag point P until you find the value A of the amplitude and the value D of the phase displacement of the function 3 cos θ + 4 sin θ. Use these values to write 3 cos θ + 4 sin θ in the form A cos(θ − D).

You can use the sketch above to approximate the amplitude and phase shift of many linear combinations of the sine and cosine functions. Now we'll investigate how the amplitude A and the phase displacement D are related to the initial parameters c and d. Keep in mind that the problem you are presented with is: Given two parameters c and d, find numbers A and D so that c cos θ + d sin θ = A cos(θ − D).

4. The first step is to examine the right-hand side of the above equation. Recall the composite argument property for cosine, cos(A − B) = cos A cos B + sin A sin B. Use the composite argument property to write the function A cos(θ − D) as an expression in sine and cosine of θ and D.
5. Use your answer to problem 4 to explain why, if c cos θ + d sin θ = A cos(θ − D), then A cos D should equal c and A sin D should equal d.

Sketch

Recall the circular function definitions of cosine and sine. Based on the results from problems 4 and 5, given two parameters c and d,, you are trying to find numbers A and D such that A cos D = c and A sin D = d. This suggests that you should look at a point in the (u, v)-plane with u-coordinate equal to c and v-coordinate equal to d.

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Investigate

6. Assuming that u = c = A cos D and v = d = A sin D, explain why the above triangle has hypotenuse A and makes angle D with the positive u-axis.
7. Set u = 3 and v = 4, and observe the values of A and D. Are these the same values of A and D that you observed when you set c = 3 and d = 4 in the first sketch?
8. Set u = −2 and v = 3 in the second sketch, and observe the values of A and D. Scroll up to the first sketch, and set c = −2 and d = 3. Make sure that the red graph of the function c cos θ + d sin θ is showing, and drag point P to the first peak to the right of the y-axis. How are the coordinates of P related to the values of A and D?
9. Use the triangle to explain why A² = c² + d² and D = arctan(d/c). Is it always true that D = tan⁻¹(d/c)?
10. Use the domain and range of the function tan⁻¹θ to explain why, if c is positive, then D = tan⁻¹(d/c) and if c is negative, then D = tan⁻¹(d/c) ± 180°.
11. Use what you have learned in this investigation to write −4 cos &theta − 5 sin &theta as a transformed cosine function.