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Trigonometric Addition Formulas — Definition, Formula & Examples

Trigonometric addition formulas express the sine, cosine, or tangent of a sum or difference of two angles in terms of the trig functions of each angle separately. They let you find exact values for angles like 75° or 15° by breaking them into angles you already know.

The trigonometric addition formulas (also called sum and difference identities) are the identities: sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta, cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta, and tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}, valid for all real values of α\alpha and β\beta where the expressions are defined.

Key Formula

\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$ $$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$$ $$\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}
Where:
  • α\alpha = First angle (in degrees or radians)
  • β\beta = Second angle (in degrees or radians)

How It Works

To use an addition formula, decompose the target angle into a sum or difference of two angles whose sine and cosine you know exactly — typically multiples of 30°, 45°, or 60°. Substitute those known values into the appropriate formula and simplify. For example, to find sin⁡75°\sin 75°, write 75°=45°+30°75° = 45° + 30° and apply the sine addition formula. These identities also serve as the foundation for deriving double-angle, half-angle, and product-to-sum identities.

Worked Example

Problem: Find the exact value of cos 75°.
Step 1: Decompose 75° into a sum of known angles.
75°=45°+30°75° = 45° + 30°
Step 2: Apply the cosine addition formula.
cos⁡75°=cos⁡45°cos⁡30°−sin⁡45°sin⁡30°\cos 75° = \cos 45°\cos 30° - \sin 45°\sin 30°
Step 3: Substitute exact values: cos 45° = sin 45° = √2/2, cos 30° = √3/2, sin 30° = 1/2.
cos⁡75°=22⋅32−22⋅12\cos 75° = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\cdot\frac{1}{2}
Step 4: Simplify.
cos⁡75°=64−24=6−24\cos 75° = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
Answer: cos⁡75°=6−24≈0.2588\cos 75° = \dfrac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588

Another Example

Problem: Find the exact value of sin 15°.
Step 1: Write 15° as a difference of known angles.
15°=45°−30°15° = 45° - 30°
Step 2: Apply the sine subtraction formula: sin(α − β) = sin α cos β − cos α sin β.
sin⁡15°=sin⁡45°cos⁡30°−cos⁡45°sin⁡30°\sin 15° = \sin 45°\cos 30° - \cos 45°\sin 30°
Step 3: Substitute known values and simplify.
sin⁡15°=22⋅32−22⋅12=6−24\sin 15° = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}
Answer: sin⁡15°=6−24≈0.2588\sin 15° = \dfrac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588

Why It Matters

These formulas are essential in precalculus and AP Calculus, where you need exact trig values and must simplify expressions before differentiating or integrating. Engineers and physicists use them constantly when combining oscillations, analyzing sound waves, or resolving forces at different angles. Mastering addition formulas also unlocks every other trig identity on a standard reference sheet, since double-angle and half-angle formulas are derived directly from them.

Common Mistakes

Mistake: Using the wrong sign in the cosine formula — writing cos(α + β) = cos α cos β + sin α sin β.
Correction: The cosine addition formula reverses the sign: cos(α + β) = cos α cos β − sin α sin β. The sign in the formula is always opposite to the sign in the argument.
Mistake: Distributing cosine or sine across a sum, e.g., writing sin(α + β) = sin α + sin β.
Correction: Trig functions are not linear. You must use the full addition formula; sin(α + β) = sin α cos β + cos α sin β.

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