Combining a Sine and a Cosine into a Single Wave
Write a sin x + b cos x as R sin(x + α), so two waves become one with a single amplitude and shift. Maximum and minimum values, and the x where they happen, then read off directly.
What a learner can do afterwards
- Write 3 sin x + 4 cos x in the form R sin(x + α)
- State the maximum value of that expression and the x where it occurs
- Solve an equation by rewriting the left side as a single wave
1 · Read
A sum like a sin x plus b cos x is one wave in disguise. Write it as R sin of x plus alpha. Then R squared equals a squared plus b squared, and tan alpha equals b over a. Two waves become one amplitude with one shift.
Take 3 sin x plus 4 cos x. Then R is the root of 9 plus 16, which is 5. The angle has tan alpha 4 over 3. So the sum is 5 sin of x plus alpha. One pair of numbers now does the work of two.
Maximum and minimum read off directly. The sine part peaks at 1, so the max is R and the min is minus R. For the example above the largest value is 5. It happens where x plus alpha equals 90 degrees.
Equations yield the same way. Divide 3 sin x plus 4 cos x equals 2.5 by 5 to get sin of x plus alpha equals 0.5. First check the right side sits within minus 1 and 1. Then solve the plain sine equation.
Find R and alpha, then read the max, the min, and the solutions off the single wave.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.