MathematicsChapter 9: Trigonometric Functions
More Max/min and A+b Cos
a + b cos(c theta + d): still [a-|b|, a+|b|].
Learning objectives
- State the main ideas and notation used in more max/min and a+b cos.
- Apply the rules of more max/min and a+b cos in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
a + b cos(c theta + d): still [a-|b|, a+|b|]. Combined a sin theta + b cos theta = R sin(theta+alpha) with R=sqrt(a^2+b^2) if the book uses that rewrite (then max R, min -R, plus a shift if a constant sits outside). If not in the book, skip the R-method. Meaning: one sine-or-cosine is enough for this unit's listed type; the R-form is a bonus if taught.
Mathematical form
a=4, b=-3, |b|=3
theta = R sin(theta+alpha) with R
b=-3
cos=1)
Where / symbols
- MORE MAX/MIN AND A+B COS
- a + b cos(c theta + d): still [a-|b|, a+|b|]. Combined a sin theta + b cos theta = R sin(theta+alpha) with R=sqrt(a^2+b^2) if the book uses that rewrite (then max R, min -R, plus a shift if a constant sits outside). If not in the book, skip the R-method. Meaning: one sine-or-cosine is enough for this unit's listed type; the R-form is a bonus if taught.
- 1. a + b COS
- Identical range rule. Peak when b cos = |b|. Same numbers a±|b|.
- 2. R-METHOD (IF IN BOOK)
- a sin + b cos = R sin(theta+phi), R=sqrt(a^2+b^2). Max R, min -R. Then add a constant outside. Not required if the paper only lists a+b sin(c theta+d).
Relationship
- optional R=sqrt(a2+b2)
Overview
Cosine form, then R if taught:
a + b COS
Identical range rule. Peak when b cos = |b|. Same numbers a±|b|.
Examples 5-4 cos: range [1,9] · a=5, |b|=4 · Write a-|b| to a+|b|
R-Method (if in Book)
a sin + b cos = R sin(theta+phi), R=sqrt(a^2+b^2). Max R, min -R. Then add a constant outside. Not required if the paper only lists a+b sin(c theta+d).
Examples R^2=a^2+b^2 · Max R · Only if taught
WHAT — MORE MAX/MIN AND A+B COS
a + b cos(c theta + d): still [a-|b|, a+|b|]. Combined a sin theta + b cos theta = R sin(theta+alpha) with R=sqrt(a^2+b^2) if the book uses that rewrite (then max R, min -R, plus a shift if a constant sits outside). If not in the book, skip the R-method. Meaning: one sine-or-cosine is enough for this unit's listed type; the R-form is a bonus if taught.
RULE — 1. a + b COS
Identical range rule. Peak when b cos = |b|. Same numbers a±|b|.
METHOD — 2. R-METHOD (IF IN BOOK)
a sin + b cos = R sin(theta+phi), R=sqrt(a^2+b^2). Max R, min -R. Then add a constant outside. Not required if the paper only lists a+b sin(c theta+d).
WORKED EXAMPLE
Worked: min of 4 - 3 cos theta. a=4, b=-3, |b|=3 min a-|b|=1 max a+|b|=7 min 1 (when -3 cos is as small as -3, i.e. cos=1)
How to answer a more max/min and a+b cos 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
- min of 4 - 3 cos theta.
- a=4, b=-3, |b|=3
- min a-|b|=1
- max a+|b|=7
- min 1 (when -3 cos is as small as -3, i.e. cos=1)
In simple words
For 4-3 cos, min is 1 not -3. Shift by 4 first in your head.
Key points
- a+b cos same as a+b sin for range
- optional R=sqrt(a2+b2)
- |b| again
- R hypotenuse
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.More Max/min and A+b Cos
Sticky Notes · Quick Revision
Core Idea
- a + b cos(c theta + d): still [a-|b|, a+|b|].
- Combined a sin theta + b cos theta = R sin(theta+alpha) with R=sqrt(a^2+b^2) if the book uses that rewrite (then max R, min -R, plus a shift if a constant sits outside).
- If not in the book, skip the R-method.
1. a + b COS
- Identical range rule.
- Peak when b cos = |b|.
- Same numbers a±|b|.
2. R-METHOD (IF IN BOOK)
- a sin + b cos = R sin(theta+phi), R=sqrt(a2+b2).
- Max R, min -R.
- Then add a constant outside.
- Not required if the paper only lists a+b sin(c theta+d).
Worked Example
- Worked: min of 4 - 3 cos theta.
- a=4, b=-3, |b|=3
- min a-|b|=1
- max a+|b|=7
- min 1 (when -3 cos is as small as -3, i.e. cos=1)
Common Misconception
- For 4-3 cos, min is 1 not -3. Shift by 4 first in your head.
FBISE Exam Check
- State the rule for more max/min and a+b cos before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- a + b cos(c theta + d): still [a-|b|, a+|b|].
- Combined a sin theta + b cos theta = R sin(theta+alpha) with R=sqrt(a^2+b^2) if the book uses that rewrite (then max R, min -R, plus a shift if a constant sits outside).
- If not in the book, skip the R-method.
- For 4-3 cos, min is 1 not -3. Shift by 4 first in your head.
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