MathematicsChapter 9: Trigonometric Functions
MAX AND MIN OF a + b sin(theta)
For real theta, sin theta runs over [-1,1], so a + b sin theta runs over [a-|b|, a+|b|].
Learning objectives
- State the main ideas and notation used in max and min of a + b sin(theta).
- Apply the rules of max and min of a + b sin(theta) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
For real theta, sin theta runs over [-1,1], so a + b sin theta runs over [a-|b|, a+|b|]. Max a+|b|, min a-|b|. Same for a + b cos. For a + b sin(c theta + d), still the same max/min (c and d do not change the range). Reciprocal: 1a+b sin needs a+b sin never 0, and the max/min swap in the way of taking reciprocals of a range away from 0 (school: if the range of the inside is [m,M] with 0 not in [m,M], range of reciprocal is [1M, 1m] or swapped if negative — treat signs carefully).
Where / symbols
- MAX AND MIN OF a + b sin(theta)
- For real theta, sin theta runs over [-1,1], so a + b sin theta runs over [a-|b|, a+|b|]. Max a+|b|, min a-|b|. Same for a + b cos. For a + b sin(c theta + d), still the same max/min (c and d do not change the range). Reciprocal: 1/(a+b sin) needs a+b sin never 0, and the max/min swap in the way of taking reciprocals of a range away from 0 (school: if the range of the inside is [m,M] with 0 not in [m,M], range of reciprocal is [1/M, 1/m] or swapped if negative — treat signs carefully).
- 1. a + b sin
- Max when sin = sign of b (i.e. b sin = |b|). Min when b sin = -|b|. State theta if asked: sin=1 or -1, then solve c theta+d.
- 2. RECIPROCAL TYPE
- If y=1/(a+b sin) and |a|>|b|, the inside never 0. Then y_max = 1/(inside_min), y_min = 1/(inside_max) when inside>0. If inside stays negative, inequalities flip. Show the inside range first.
Overview
Bounded sine, then reciprocal:
a + b sin
Max when sin = sign of b (i.e. b sin = |b|). Min when b sin = -|b|. State theta if asked: sin=1 or -1, then solve c theta+d.
Examples a+3 sin: max a+3 · a-2 cos: |b|=2 · Same for cos
Reciprocal Type
If y=1a+b sin and |a|>|b|, the inside never 0. Then ymax = 1insidemin, ymin = 1insidemax when inside>0. If inside stays negative, inequalities flip. Show the inside range first.
Examples |a|>|b| for all-real domain · Reciprocal swaps max/min if positive · State domain holes if |a|<|b|
WHAT — MAX AND MIN OF a + b sin(theta)
For real theta, sin theta runs over [-1,1], so a + b sin theta runs over [a-|b|, a+|b|]. Max a+|b|, min a-|b|. Same for a + b cos. For a + b sin(c theta + d), still the same max/min (c and d do not change the range). Reciprocal: 1a+b sin needs a+b sin never 0, and the max/min swap in the way of taking reciprocals of a range away from 0 (school: if the range of the inside is [m,M] with 0 not in [m,M], range of reciprocal is [1M, 1m] or swapped if negative — treat signs carefully).
RULE — 1. a + b sin
Max when sin = sign of b (i.e. b sin = |b|). Min when b sin = -|b|. State theta if asked: sin=1 or -1, then solve c theta+d.
METHOD — 2. RECIPROCAL TYPE
If y=1a+b sin and |a|>|b|, the inside never 0. Then ymax = 1insidemin, ymin = 1insidemax when inside>0. If inside stays negative, inequalities flip. Show the inside range first.
WORKED EXAMPLE
Worked: max of 5 + 2 sin(3x+1). 2 sin between -2 and 2 so 5+2 sin between 3 and 7 max 7, min 3 3x+1 does not change that
How to answer a max and min of a + b sin(theta) question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- max of 5 + 2 sin(3x+1).
- 2 sin between -2 and 2
- so 5+2 sin between 3 and 7
- max 7, min 3
- 3x+1 does not change that
In simple words
The 3 in sin(3x) does not change max/min. It only changes period.
Key points
- max a+|b|, min a-|b|
- c and d do not change those
- |b| stretch
- inside never 0
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.MAX AND MIN OF a + b sin(theta)
Sticky Notes · Quick Revision
Core Idea
- For real theta, sin theta runs over [-1,1], so a + b sin theta runs over [a-|b|, a+|b|].
- Max a+|b|, min a-|b|.
- Same for a + b cos.
1. a + b sin
- Max when sin = sign of b (i.e.
- b sin = |b|).
- Min when b sin = -|b|.
- State theta if asked: sin=1 or -1, then solve c theta+d.
2. RECIPROCAL TYPE
- If y=1a+b sin and |a|>|b|, the inside never 0.
- Then ymax = 1insidemin, ymin = 1insidemax when inside>0.
- If inside stays negative, inequalities flip.
- Show the inside range first.
Worked Example
- Worked: max of 5 + 2 sin(3x+1).
- 2 sin between -2 and 2
- so 5+2 sin between 3 and 7
- max 7, min 3
- 3x+1 does not change that
Common Misconception
- The 3 in sin(3x) does not change max/min. It only changes period.
FBISE Exam Check
- State the rule for max and min of a + b sin(theta) before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- For real theta, sin theta runs over [-1,1], so a + b sin theta runs over [a-|b|, a+|b|].
- Max a+|b|, min a-|b|.
- Same for a + b cos.
- The 3 in sin(3x) does not change max/min. It only changes period.
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