R-form turns a combined sine-cosine expression into one sinusoid whose range can be read directly. This makes many maximum and minimum problems shorter, but the unrestricted range must not be confused with the range over a restricted interval.
1. The basic range result
If a sin x+b cos x=Rsin(x+α), then −1≤sin(x+α)≤1. Therefore
−R≤a sin x+b cos x≤R, where R=√(a²+b²).
2. Worked example: unrestricted extrema
Find the maximum and minimum of 3sin x+4cos x.
R=5, so the expression can be written 5sin(x+α). Its unrestricted maximum is 5 and minimum is −5.
You do not need the numerical value of α if the question asks only for the unrestricted extreme values.
3. Add a vertical translation
For 3sin x+4cos x+7, the oscillating part ranges from −5 to 5. Adding 7 gives
2≤3sin x+4cos x+7≤12.
The maximum is 12 and minimum is 2.
4. Multipliers outside the combined expression
If F(x)=2[3sin x+4cos x]−1, then the bracket ranges from −5 to 5. Multiplying by 2 gives −10 to 10, then subtracting 1 gives −11≤F(x)≤9.
5. When does the maximum occur?
If 3sin x+4cos x=5sin(x+α), the maximum occurs when sin(x+α)=1.
Thus x+α=π/2+2πn, so x=π/2−α+2πn.
The minimum occurs when sin(x+α)=−1, giving x=3π/2−α+2πn.
6. Restricted intervals require an extra check
The unrestricted maximum R is attained only if the allowed interval contains an x for which the shifted sine reaches 1.
For a restricted interval, find candidate interior extrema from the R-form condition and compare them with endpoint values. Do not report ±R automatically.
7. Worked restricted-interval example
Consider F(x)=sin x+cos x on 0≤x≤π/6.
R=√2 and F(x)=√2 sin(x+π/4). The unrestricted maximum √2 would occur at x=π/4, which lies outside the interval.
On the stated interval the shifted angle runs from π/4 to 5π/12, where sine is increasing. Hence the minimum is F(0)=1 and maximum is F(π/6)=1/2+√3/2=(1+√3)/2.
8. Range can test whether an equation is solvable
Since 5sin x+12cos x has R=13, it can never equal 14 for real x. R-form can therefore answer an existence question before any inverse trigonometric calculation.
9. A modelling interpretation
Suppose a hypothetical periodic model is H(t)=20+6sin t+8cos t. The oscillating component has amplitude 10, so the model ranges from 10 to 30 over unrestricted t.
This statement describes the mathematical model. Whether a real system actually attains those values depends on whether the model and its time domain are appropriate.
10. Exact forms improve checking
For coefficients 1 and 1, R=√2. Keeping √2 exact preserves the relationship. Replacing it immediately with 1.414 can make later exact comparisons less transparent.
11. Common mistakes
- Using |a|+|b| instead of √(a²+b²).
- Forgetting a vertical translation when reporting the range.
- Assuming the unrestricted extrema occur inside every restricted interval.
- Finding the maximum value but not the x-value when both are requested.
- Rounding the phase before checking an interval boundary.
12. Practice
- Find the unrestricted range of 8sin x+15cos x.
- Find the maximum and minimum of 5+5sin x+12cos x.
- Can 7sin x+24cos x equal 26 for real x?
- If 3sin x+4cos x=5sin(x+α), state a general condition for its maximum.
- Find the unrestricted range of 2−4sin x+3cos x.
13. Answers
- R=17, so −17≤expression≤17.
- R=13, so range −8 to 18; minimum −8, maximum 18.
- No. R=25, so the largest possible value is 25.
- sin(x+α)=1, so x+α=π/2+2πn.
- R=5 for the oscillating part, so −3≤2−4sin x+3cos x≤7.
A strong checking habit
After finding a claimed maximum, compare it with the simple bound |a sin x+b cos x|≤√(a²+b²). If the result exceeds that amplitude before any vertical translation, something is wrong.
Continue with Deriving R-Form Step by Step, R-Form in Additional Mathematics, or return to the Additional Mathematics Hub.