Combine waves and convert trigonometric sums and products

Combine terms with the same frequency

For real coefficients not both zero, expand the displayed single-wave form and match coefficients. Use a positive amplitude, then select the phase from zero inclusive to two pi exclusive using both signed ratios

For 3 sin x−4 cos x=R sin(x+φ), with R>0, find R.

For 3 sin x−4 cos x=5 sin(x+φ), find sinφ

For 3 sin x−4 cos x=5 sin(x+φ), find cosφ

For 3 sin x−4 cos x=5 sin(x+φ), which quadrant contains φ?

A cosine form matches different coefficients

For a sin x+b cos x=R cos(x−φ), expansion gives R sinφ=a and R cosφ=b. State the chosen form before calculating the phase. The same function has several valid descriptions unless amplitude sign and a phase interval are fixed

Write −5 sin x+12 cos x=R cos(x−φ). Find R.

For −5 sin x+12 cos x=13 cos(x−φ), find sinφ

Axis and zero cases

If one coefficient is zero, use the unit-circle axes directly: for example, 4 cos x=4 sin(x+π/2). If both coefficients are zero, the function is identically zero; its amplitude is zero and no unique phase exists. There is no positive fundamental period for a constant function

Express 4 cos x as 4 sin(x+φ) with 0≤φ<2π. Give φ/π

Derive the amplitude and phase, ungraded

Starting from the sine compound formula, prove the amplitude-phase representation for all real a,b not both zero. Explain how to select the phase, including axis cases, and why the zero function needs separate treatment. The proof is ungraded

Model: match two unit-circle coordinates

Set R=√(a²+b²)>0. Then (a/R)²+(b/R)²=1, so these are the cosine and sine of some φ, unique modulo 2π. The compound formula gives R sin(x+φ)=a sin x+b cos x. Both signs select the quadrant and zero coordinates select an axis. For a=b=0 division by R is unavailable; every phase would describe the zero wave if amplitude zero were permitted

Read features after combining

For f(x)=a sin(nx)+b cos(nx)+k with n>0 and x real, combine to R sin(nx+φ)+k. Its range is [k−R,k+R] and its period is 2π/n. A sine-form maximum occurs when nx+φ=π/2+2jπ; dividing by n gives the corresponding x-values

Find the amplitude of f(x)=−3 sin2x+4 cos2x+10.

Find the global maximum of f(x)=−3 sin2x+4 cos2x+10.

For f(x)=−3 sin2x+4 cos2x+10, give the coefficient of π in its fundamental period

Turn sums into products

Set u=(A+B)/2 and v=(A−B)/2, so A=u+v and B=u−v. Adding or subtracting the compound formulas gives the four sum-to-product identities. Keep the original order in a difference

Write sin4x+sin x=2 sin(Px)cos(Qx), with P=(4+1)/2. Find P.

In sin4x+sin x=2 sin(5x/2)cos(Qx), with Q>0, find Q.

Write cos5x−cos x=C sin3x sin2x. Find the signed factor C.

Turn products into sums

Add or subtract compound identities and divide by two. Notice that sin A sin B produces a difference of cosines; the cos(A+B) term has a negative coefficient

In sin5x cos2x=U sin7x+V sin3x, find U.

In sin5x sin2x=U cos7x+V cos3x, find U.

Order matters when sine is involved

For cos A sin B, interchange the sine and cosine factors first: sin B cos A=½[sin(B+A)+sin(B−A)]. If rewriting sin(B−A) as sin(A−B), also reverse its sign. Cosine is even, so cos(B−A)=cos(A−B) without a sign change

Derive and check the conversion laws, ungraded

Derive the four sum-to-product formulas using u,v, and the three product-to-sum formulas directly from compound identities. Explain the negative sign for cosine difference and sine products. These derivations are ungraded

Model: add and subtract systematically

Adding sin(u+v) and sin(u−v) gives 2sin u cos v; subtracting gives 2cos u sin v. Adding the cosine expansions gives 2cos u cos v; subtracting gives −2sin u sin v. For product-to-sum, add sin(A+B) and sin(A−B) to obtain 2 sin A cos B. Add cos(A+B) and cos(A−B) to obtain 2 cos A cos B. Subtract cos(A+B) from cos(A−B) to obtain 2 sin A sin B. These identities use no division by a trig ratio, so they hold for every real A,B.

Choose a form to solve an equation, ungraded

Solve √3 sin x+cos x=1 and sin3x+sin x=0, each on 0≤x<2π. Use a suitable equivalent form and check endpoints. Independent equation solutions are ungraded

Model: shift one wave, factor the other

The first expression is 2sin(x+π/6), so sin(x+π/6)=1/2. The solutions in the specified interval are x=0 and 2π/3. The second is 2sin2x cos x=0, giving x=0,π/2,π,3π/2. The cos x=0 roots overlap the sin2x=0 roots and are listed once; 2π is excluded. Dividing by cos x would lose valid roots

Apply and assess a periodic model, ungraded

Suppose a device is modelled by y(t)=10−3sin2t+4cos2t for t in seconds. Derive its range and period, and use a cosine form to identify its maximum times. Explain which physical claims still need evidence. This modelling and explanation are ungraded

Model: combine a common angular frequency

The period is measured in seconds; retain nonnegative maximum times when required. Whether the device actually stays periodic, has those coefficients or remains in that range must be tested against observations; the algebra alone does not validate the model