8.1 Allied Angles and Sum/Difference Formulas | Fundamentals Of Trigonometry | Mathematics 11

Angle Measures and Allied Angles

Two angles are called allied angles if their sum or difference is a multiple of $9 0^{\circ}$ (or $\frac{π}{2}$ ). The trigonometric ratios of allied angles can be expressed in terms of the ratios of the original angle.

Allied Angle	sin	cos	tan
$- θ$	$- sin θ$	$cos θ$	$- tan θ$
$9 0^{\circ} - θ$	$cos θ$	$sin θ$	$cot θ$
$9 0^{\circ} + θ$	$cos θ$	$- sin θ$	$- cot θ$
$18 0^{\circ} - θ$	$sin θ$	$- cos θ$	$- tan θ$
$18 0^{\circ} + θ$	$- sin θ$	$- cos θ$	$tan θ$
$27 0^{\circ} - θ$	$- cos θ$	$- sin θ$	$cot θ$
$27 0^{\circ} + θ$	$- cos θ$	$sin θ$	$- cot θ$
$36 0^{\circ} - θ$	$- sin θ$	$cos θ$	$- tan θ$

Memory Aid: For multiples of $9 0^{\circ}$ , the function name changes (sin ↔ cos, tan ↔ cot). For multiples of $18 0^{\circ}$ , the function name stays the same. The sign is determined by the CAST rule for the quadrant.

Fundamental Law of Trigonometry (Sum/Difference Formulas)

These are the core identities derived from the fundamental law:

Sine Formulas

$sin (α + β) = sin α cos β + cos α sin β$ $sin (α - β) = sin α cos β - cos α sin β$

Cosine Formulas

$cos (α + β) = cos α cos β - sin α sin β$ $cos (α - β) = cos α cos β + sin α sin β$

Key observation: In the cosine formulas, the sign is opposite to the operator between the angles.

Tangent Formulas

$tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β}$ $tan (α - β) = \frac{t a n α - t a n β}{1 + t a n α t a n β}$

Finding Missing Ratios Using Quadrant Information

When applying sum/difference formulas, you often need to find a missing ratio (e.g., $cos α$ when $sin α$ is given).

Method:

Use the Pythagorean identity: $sin^{2} α + cos^{2} α = 1$
Solve: $cos α = \pm 1 - sin^{2} α$
Determine the correct sign from the quadrant of $α$ :
- Quadrant I: all ratios positive
- Quadrant II: sin positive, cos and tan negative
- Quadrant III: tan positive, sin and cos negative
- Quadrant IV: cos positive, sin and tan negative

Example: If $sin α = \frac{3}{5}$ and $α$ is in Quadrant II, find $cos α$ . $cos α = - 1 - (\frac{3}{5})^{2} = - 1 - \frac{9}{25} = - \frac{16}{25} = - \frac{4}{5}$

Finding Exact Values of Trigonometric Functions

Sum/difference formulas allow us to find exact values for non-standard angles by expressing them as sums or differences of standard angles ( $3 0^{\circ}, 4 5^{\circ}, 6 0^{\circ}$ , etc.).

Example: Find the exact value of $cos 1 5^{\circ}$ . $cos 1 5^{\circ} = cos (4 5^{\circ} - 3 0^{\circ})$ $= cos 4 5^{\circ} cos 3 0^{\circ} + sin 4 5^{\circ} sin 3 0^{\circ}$ $= \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}$ $= \frac{6}{4} + \frac{2}{4} = \frac{6 + 2}{4}$

Simplifying Trigonometric Expressions

Recognising the pattern of sum/difference formulas allows complex expressions to be collapsed into a single function.

Example: Simplify $\frac{c o s γ + s i n γ}{c o s γ - s i n γ}$ .

Divide numerator and denominator by $cos γ$ : $= \frac{1 + t a n γ}{1 - t a n γ} = \frac{t a n \frac{π}{4} + t a n γ}{1 - t a n \frac{π}{4} t a n γ} = tan (\frac{π}{4} + γ)$

Angle Measures and Allied Angles

Allied Angle	sin	cos	tan
$- θ$	$- sin θ$	$cos θ$	$- tan θ$
$9 0^{\circ} - θ$	$cos θ$	$sin θ$	$cot θ$
$9 0^{\circ} + θ$	$cos θ$	$- sin θ$	$- cot θ$
$18 0^{\circ} - θ$	$sin θ$	$- cos θ$	$- tan θ$
$18 0^{\circ} + θ$	$- sin θ$	$- cos θ$	$tan θ$
$27 0^{\circ} - θ$	$- cos θ$	$- sin θ$	$cot θ$
$27 0^{\circ} + θ$	$- cos θ$	$sin θ$	$- cot θ$
$36 0^{\circ} - θ$	$- sin θ$	$cos θ$	$- tan θ$

Fundamental Law of Trigonometry (Sum/Difference Formulas)

These are the core identities derived from the fundamental law:

Sine Formulas

$sin (α + β) = sin α cos β + cos α sin β$ $sin (α - β) = sin α cos β - cos α sin β$

Cosine Formulas

$cos (α + β) = cos α cos β - sin α sin β$ $cos (α - β) = cos α cos β + sin α sin β$

Key observation: In the cosine formulas, the sign is opposite to the operator between the angles.

Tangent Formulas

$tan (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β}$ $tan (α - β) = \frac{t a n α - t a n β}{1 + t a n α t a n β}$

Finding Missing Ratios Using Quadrant Information

When applying sum/difference formulas, you often need to find a missing ratio (e.g., $cos α$ when $sin α$ is given).

Method:

Use the Pythagorean identity: $sin^{2} α + cos^{2} α = 1$
Solve: $cos α = \pm 1 - sin^{2} α$
Determine the correct sign from the quadrant of $α$ :
- Quadrant I: all ratios positive
- Quadrant II: sin positive, cos and tan negative
- Quadrant III: tan positive, sin and cos negative
- Quadrant IV: cos positive, sin and tan negative

Example: If $sin α = \frac{3}{5}$ and $α$ is in Quadrant II, find $cos α$ . $cos α = - 1 - (\frac{3}{5})^{2} = - 1 - \frac{9}{25} = - \frac{16}{25} = - \frac{4}{5}$

Finding Exact Values of Trigonometric Functions

Sum/difference formulas allow us to find exact values for non-standard angles by expressing them as sums or differences of standard angles ( $3 0^{\circ}, 4 5^{\circ}, 6 0^{\circ}$ , etc.).

Simplifying Trigonometric Expressions

Recognising the pattern of sum/difference formulas allows complex expressions to be collapsed into a single function.

Example: Simplify $\frac{c o s γ + s i n γ}{c o s γ - s i n γ}$ .

Divide numerator and denominator by $cos γ$ : $= \frac{1 + t a n γ}{1 - t a n γ} = \frac{t a n \frac{π}{4} + t a n γ}{1 - t a n \frac{π}{4} t a n γ} = tan (\frac{π}{4} + γ)$