➕ Trigonometric Addition Formulas
Sum and Difference Identities for Sine, Cosine, Tangent & Cotangent
Sum and Difference Formulas
These formulas express trigonometric functions of sums or differences of angles in terms of functions of the individual angles.
Sine Addition Formulas
Formulas for \( \sin(x \pm y) \)
Sine of Sum
\[ \sin(x + y) = \sin x \cos y + \cos x \sin y \]
📝 Memory Aid:
“Sine Cosine, Cosine Sine, Keep the Sign”
(sin cos + cos sin)
Sine of Difference
\[ \sin(x – y) = \sin x \cos y – \cos x \sin y \]
📝 Memory Aid:
Same as sum but with minus sign:
(sin cos – cos sin)
🎯 Example: Calculate \( \sin(75^\circ) \)
Using \( 75^\circ = 45^\circ + 30^\circ \):
\( \sin(75^\circ) = \sin(45^\circ + 30^\circ) \)
\( = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \)
\( = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \)
\( = \frac{\sqrt{6} + \sqrt{2}}{4} \)
Cosine Addition Formulas
Formulas for \( \cos(x \pm y) \)
Cosine of Sum
\[ \cos(x + y) = \cos x \cos y – \sin x \sin y \]
📝 Memory Aid:
“Cosine Cosine, Sine Sine, Change the Sign”
(cos cos – sin sin)
Cosine of Difference
\[ \cos(x – y) = \cos x \cos y + \sin x \sin y \]
📝 Memory Aid:
Same as sum but with plus sign:
(cos cos + sin sin)
🎯 Example: Calculate \( \cos(15^\circ) \)
Using \( 15^\circ = 45^\circ – 30^\circ \):
\( \cos(15^\circ) = \cos(45^\circ – 30^\circ) \)
\( = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \)
\( = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \)
\( = \frac{\sqrt{6} + \sqrt{2}}{4} \)
Tangent Addition Formulas
Formulas for \( \tan(x \pm y) \)
Tangent of Sum
\[ \tan(x + y) = \frac{\tan x + \tan y}{1 – \tan x \tan y} \]
📝 Memory Aid:
Numerator: Sum of tangents
Denominator: 1 minus product of tangents
Tangent of Difference
\[ \tan(x – y) = \frac{\tan x – \tan y}{1 + \tan x \tan y} \]
📝 Memory Aid:
Numerator: Difference of tangents
Denominator: 1 plus product of tangents
🎯 Example: Calculate \( \tan(75^\circ) \)
Using \( 75^\circ = 45^\circ + 30^\circ \) and \( \tan 45^\circ = 1 \), \( \tan 30^\circ = \frac{1}{\sqrt{3}} \):
\( \tan(75^\circ) = \frac{1 + \frac{1}{\sqrt{3}}}{1 – 1 \cdot \frac{1}{\sqrt{3}}} \)
\( = \frac{\sqrt{3} + 1}{\sqrt{3} – 1} \)
\( = 2 + \sqrt{3} \)
Cotangent Addition Formulas
Formulas for \( \cot(x \pm y) \)
Cotangent of Sum
\[ \cot(x + y) = \frac{\cot x \cot y – 1}{\cot y + \cot x} \]
📝 Memory Aid:
Numerator: Product minus 1
Denominator: Sum of cotangents
Cotangent of Difference
\[ \cot(x – y) = \frac{\cot x \cot y + 1}{\cot y – \cot x} \]
📝 Memory Aid:
Numerator: Product plus 1
Denominator: Difference of cotangents
📝 Alternative Derivation:
Cotangent formulas can also be derived from tangent formulas:
Since \( \cot \theta = \frac{1}{\tan \theta} \)