The R cos(θ ± α) Form

SPEC 5.6 - TRIGONOMETRIC IDENTITIES

Combining trigonometric functions into a single expression

Why R-Form?

Expressions like

a cos θ + b sin θ

appear frequently in mathematics. The R-form allows us to write these as a single trigonometric function.

Key Idea

Combining

a cos θ + b sin θ

into the form

R cos (θ - α)

makes it easier to:

Find maximum and minimum values

Determine when these occur

Sketch graphs

The Key Results

Cosine Forms

a cos θ + b sin θ = R cos (θ - α)

a cos θ - b sin θ = R cos (θ + α)

Sine Forms

a sin θ + b cos θ = R sin (θ + α)

a sin θ - b cos θ = R sin (θ - α)

Memory Tip

Cosine forms: sign in bracket is opposite to sign between terms

Sine forms: sign in bracket is the same as sign between terms

Finding R and α

Key Formulae

R = a^{2} + b^{2}

tan α = \frac{b}{a}

Important

R is always positive

α is acute:

0° < α < 90°

or

0 < α < \frac{π}{2}

Express α in degrees or radians depending on the question

Worked Example01

Express $3 cos θ + 4 sin θ$ in the form $R cos (θ - α)$ , where $R > 0$ and $0° < α < 90°$ .

1

Find R:

R = a^{2} + b^{2} = 3^{2} + 4^{2} = 9 + 16 = 25 = 5

2

Find α:

tan α = \frac{b}{a} = \frac{4}{3}

3

α = tan^{- 1} (\frac{4}{3}) = 53.13°

(2 d.p.)

Answer

3 cos θ + 4 sin θ = 5 cos (θ - 53.13°)

Worked Example02

Express $5 sin x - 12 cos x$ in the form $R sin (x - α)$ , where $R > 0$ and $0 < α < \frac{π}{2}$ .

1

Find R:

R = 5^{2} + 1 2^{2} = 25 + 144 = 169 = 13

2

Find α:

tan α = \frac{12}{5}

3

α = tan^{- 1} (\frac{12}{5}) = 1.176

radians (3 d.p.)

Answer

5 sin x - 12 cos x = 13 sin (x - 1.176)

Practice

Question: Express

3 cos θ + sin θ

in the form

R cos (θ - α)

, where

R > 0

and

0° < α < 90°

.

1

R = (3)^{2} + 1^{2} = 3 + 1 = 4 = 2

2

tan α = \frac{1}{3}

, so

α = 30°

Answer

3 cos θ + sin θ = 2 cos (θ - 30°)

Maximum and Minimum Values

Once in R-form, finding maximum and minimum values is straightforward.

For y = R cos(θ − α)

ValueResultOccurs whenBecauseMaximum

R

θ = α

cos (0) = 1

Minimum

- R

θ = α + 180°

or

θ = α + π

cos (180°) = - 1

Method

Set the bracket

(θ - α)

equal to 0° for maximum, and 180° for minimum, then solve for θ.

Worked Example03

Find the maximum and minimum values of $3 cos θ + 4 sin θ$ and the values of θ where they occur for $0° \leq θ \leq 360°$ .

1

From Example 1:

3 cos θ + 4 sin θ = 5 cos (θ - 53.13°)

2

Maximum value = R = 5, occurs when

θ - 53.13° = 0°

, so

θ = 53.13°

3

Minimum value = -R = -5, occurs when

θ - 53.13° = 180°

, so

θ = 233.13°

AnswerMaximum = 5 at θ = 53.13°; Minimum = -5 at θ = 233.13°

Practice

Question: Find the maximum and minimum values of

5 cos θ + 12 sin θ

and the values of θ where they occur for

0° \leq θ \leq 360°

.

1

R = 25 + 144 = 13

,

tan α = \frac{12}{5}

,

α = 67.38°

2

So

5 cos θ + 12 sin θ = 13 cos (θ - 67.38°)

3

Maximum = 13 at θ = 67.38°; Minimum = -13 at θ = 247.38°

Sketching Graphs

The R-form reveals key features for sketching:

Graph Features for y = R cos(θ − α)

FeatureValueMaximum value

R

Minimum value

- R

Phase shift

α

to the rightPeriod360° or

2 π

(unchanged)Maximum point

(α, R)

Minimum point

(α + 180°, - R)

Worked Example04

Sketch the graph of $y = 3 cos θ + 4 sin θ$ for $0° \leq θ \leq 360°$ .

1

R-form:

y = 5 cos (θ - 53.13°)

2

Maximum value = 5, Minimum value = -5, Phase shift = 53.13° to the right

3

Key points: Maximum (53.13°, 5), crosses x-axis at 143.13° and 323.13°, Minimum (233.13°, -5)

Sketch: A cosine curve with max 5 and min -5, shifted 53.13° to the right

Practice

Question: Sketch the graph of

y = 3 sin x + cos x

for

0 \leq x \leq 2 π

, showing maximum and minimum points.

1

R = 3 + 1 = 2

,

tan α = \frac{1}{3}

,

α = \frac{π}{6}

2

R-form:

y = 2 sin (x + \frac{π}{6})

3

Maximum = 2 at

x = \frac{π}{3}

; Minimum = -2 at

x = \frac{4 π}{3}

Summary

Key Formulae

ExpressionR-Form

a cos θ + b sin θ

R cos (θ - α)

a cos θ - b sin θ

R cos (θ + α)

a sin θ + b cos θ

R sin (θ + α)

a sin θ - b cos θ

R sin (θ - α)

Key Results

R = a^{2} + b^{2}

and

tan α = \frac{b}{a}

α can be in degrees or radians (check the question)

Maximum = R, Minimum = -R

Graph: max/min values are ±R, phase shift α