Ratios & Identities: Trigonometric Ratios of Sum & Difference of Two Angles

Trigonometric ratios of the sum and difference of two angles are fundamental concepts in trigonometry. They allow us to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. These identities are particularly useful in solving trigonometric equations, proving other identities, and simplifying expressions.

Trigonometric Identities for Sum and Difference of Angles

The basic trigonometric identities for the sum and difference of two angles are as follows:

Sine of Sum and Difference

$$ \sin(A + B) = \sin A \cos B + \cos A \sin B $$

$$ \sin(A - B) = \sin A \cos B - \cos A \sin B $$

Cosine of Sum and Difference

$$ \cos(A + B) = \cos A \cos B - \sin A \sin B $$

$$ \cos(A - B) = \cos A \cos B + \sin A \sin B $$

Tangent of Sum and Difference

$$ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$

$$ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} $$

Important Points and Differences

Identity	Sum Formula	Difference Formula
Sine	$\sin(A + B) = \sin A \cos B + \cos A \sin B$	$\sin(A - B) = \sin A \cos B - \cos A \sin B$
Cosine	$\cos(A + B) = \cos A \cos B - \sin A \sin B$	$\cos(A - B) = \cos A \cos B + \sin A \sin B$
Tangent	$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$	$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

• The sine of the sum of two angles is the sum of the product of the sine and cosine of the individual angles, whereas the sine of the difference is the difference of the same products.
• The cosine of the sum of two angles is the difference of the product of the cosines and the product of the sines of the individual angles, whereas the cosine of the difference is the sum of these products.
• The tangent of the sum or difference of two angles is a fraction that involves the sum or difference of the tangents of the individual angles in the numerator and a term involving the product of the tangents in the denominator.

Examples

Example 1: Sine of Sum

Calculate $\sin(75^\circ)$ using the sum identity, given that $75^\circ = 45^\circ + 30^\circ$.

$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) $$

Using the identity:

$$ \sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) $$

Substitute known values:

$$ \sin(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$

$$ \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Example 2: Cosine of Difference

Calculate $\cos(15^\circ)$ using the difference identity, given that $15^\circ = 45^\circ - 30^\circ$.

$$ \cos(15^\circ) = \cos(45^\circ - 30^\circ) $$

Using the identity:

$$ \cos(15^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ) $$

Substitute known values:

$$ \cos(15^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) $$

$$ \cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Example 3: Tangent of Sum

Calculate $\tan(75^\circ)$ using the sum identity, given that $75^\circ = 45^\circ + 30^\circ$.

$$ \tan(75^\circ) = \tan(45^\circ + 30^\circ) $$

Using the identity:

$$ \tan(75^\circ) = \frac{\tan(45^\circ) + \tan(30^\circ)}{1 - \tan(45^\circ)\tan(30^\circ)} $$

Substitute known values:

$$ \tan(75^\circ) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} $$

$$ \tan(75^\circ) = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} $$

$$ \tan(75^\circ) = \frac{3 + 2\sqrt{3} + 1}{2} $$

$$ \tan(75^\circ) = 2 + \sqrt{3} $$

These examples illustrate how to use the sum and difference identities to calculate the trigonometric ratios of angles that can be expressed as the sum or difference of more easily manageable angles. Mastery of these identities is crucial for success in trigonometry and related fields such as physics, engineering, and computer science.