--- title: "Combining a sine and a cosine into a single wave" description: "For real a and b, not both zero, write a sin x + b cos x as R sin(x + α) with R > 0. Choose α from both coefficient signs. Use the single-wave form to find its range, locate maxima and minima, and sol" canonical: https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq source: https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Combining a sine and a cosine into a single wave For real a and b, not both zero, write a sin x + b cos x as R sin(x + α) with R > 0. Choose α from both coefficient signs. Use the single-wave form to find its range, locate maxima and minima, and solve equations on a stated interval. Subject: Mathematics · Area: Geometry · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq ## Ready when they can - Combine same-angle sine and cosine terms by matching coefficients, including zero or negative coefficients, and check a proposed form - Find the range and locations of maxima and minima of a combined wave, including a constant offset or a stated x interval - Solve single-wave equations on a stated interval, checking the sine range and keeping all valid solution branches ## Lesson: Two waves in, one wave out As with double-angle identities, start from the sine addition rule. Work in degrees here. Expand R sin(x + α) = R cos α sin x + R sin α cos x. To match a sin x + b cos x, choose R cos α = a and R sin α = b. Square and add: R²(cos²α + sin²α) = a² + b². Since the bracket is 1, choose R = √(a² + b²) > 0 when the coefficients are not both zero. *(drawing: R cos α is the sine coefficient. R sin α is the cosine coefficient.)* For a sin x + b cos x with R > 0, choose cos α = a/R and sin α = b/R. Their signs select the quadrant. If a ≠ 0, tan α = b/a, but inverse tangent alone may select the other quadrant. Choose 0° ≤ α < 360°, unless another interval is stated. Since sin 45° = cos 45° = 1/√2, equal positive values give α = 45°; a negative cosine value and a positive sine value of equal magnitude give α = 135°. If a = 0 and b > 0, choose α = 90°; if a = 0 and b < 0, choose α = 270°. If a = b = 0, use R = 0 and do not divide by R. Both terms must use the same angle for this coefficient-matching rule. A shifted sine changes sign when x increases by 180°, since sin(y + 180°) = -sin y. An identity must hold at every x, so one unequal pair of values disproves it. **Example.** For 7 sin x + 24 cos x, R = √(49 + 576) = 25. Matching gives cos α = 7/25 and sin α = 24/25. Both are positive, so α is acute. Thus tan α = 24/7 and α ≈ 73.7°. The expression is 25 sin(x + α). Keep the unrounded angle for later calculations. *(drawing: Sine reaches both endpoints and every value between them.)* For R > 0 and unrestricted real x, the range is -R to R. Maxima occur at x + α = 90° + 360°n; minima occur at 270° + 360°n, for integer n. Adding c shifts the range to c - R through c + R. On a restricted interval, check allowed peaks, troughs and endpoints. Sine increases from 0° to 90°. To solve c + R sin(x + α) = d with R > 0, subtract c and divide by R: sin(x + α) = k, where k = (d - c)/R. If |k| > 1, no real solution exists. Otherwise set β = arcsin k, with -90° ≤ β ≤ 90°. For y = x + α, the solutions are y = β + 360°n or y = 180° - β + 360°n, for integer n. Subtract α, keep only x in the stated interval and remove repeated answers. **Example.** Solve 6 sin(x + 20°) = 3 for 0° ≤ x < 360°. Divide by 6 to get sin y = 0.5, where y = x + 20°. The shifted interval is 20° ≤ y < 380°. The sine solutions in it are 30° and 150°. Subtract 20° to get x = 10° and 130°. Both answers lie in the stated x interval. *(drawing: Shift the interval with the angle, then shift the answers back.)* **Recap.** Match coefficients to find R and the phase signs. Use the sine range for extrema. For equations, find both angle families, subtract the phase and check the interval. *This lesson, practice bank and flashcards were written for this dataset.* ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Double Angle Formulae](https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam) ## Optional connections - [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho) ## Opens up - [Phasor Addition and the Mathematics of Interference](https://lightmysky.com/learn/science/phasor-addition-and-the-mathematics-of-interference-mt_dxh7HHbIUw) - [Impedance and Phasors in AC Circuits](https://lightmysky.com/learn/science/impedance-and-phasors-in-ac-circuits-mt_vQVlVWv2ln) - [Damped and Driven Oscillations](https://lightmysky.com/learn/mathematics/damped-and-driven-oscillations-mt_Y8K1njb1PH)