MAX AND MIN OF a + b sin(theta)

MAX AND MIN OF a + b sin(theta)

What is it?

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).

In simple words

• 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.

Important terms

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).

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 = 1inside_min, ymin = 1inside_max when inside>0. If inside stays negative, inequalities flip. Show the inside range first.

What you should learn

• 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.

Examples

• a+3 sin: max a+3
• a-2 cos: |b|=2
• Same for cos

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.

RECIPROCAL TYPE

If y=1a+b sin and |a|>|b|, the inside never 0. Then ymax = 1inside_min, ymin = 1inside_max when inside>0. If inside stays negative, inequalities flip. Show the inside range first.

Full school-level explanation

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).

Bounded sine, then reciprocal:

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 = 1inside_min, ymin = 1inside_max 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

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.

Remember

• |b| stretch

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 = 1inside_min, ymin = 1inside_max when inside>0. If inside stays negative, inequalities flip. Show the inside range first.

Remember

• inside never 0

Examples

• |a|>|b| for all-real domain
• Reciprocal swaps max/min if positive
• State domain holes if |a|<|b|

How to answer a max and min of a + b sin(theta) question

How it works

• 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.

Remember

• max a+|b|, min a-|b|
• c and d do not change those
• max 7
• c is not amplitude
• |b| stretch
• inside never 0
• max a+|b|, min a-|b|
• c and d do not change those

How the ideas connect

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

Deeper understanding

Go beyond memorising MAX AND MIN OF a + b sin(theta). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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 mistakes

Exam focus

Core Idea

1. a + b sin

2. RECIPROCAL TYPE

Worked Example

Exam focus

Quick revision

• max a+|b|, min a-|b|
• c and d do not change those
• max 7
• c is not amplitude
• 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|,
• 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 -

The 3 in sin(3x) does not change max/min. It only changes period.

For long answers on MAX AND MIN OF a + b sin(theta): define clearly → explain the main parts → give one example → state one exam caution.