Understanding Trigonometric Ratios of Compound Angles

Trigonometric ratios of compound angles describe the functional values of sine, cosine, and tangent for angles formed by the sum or difference of two arbitrary angles. These relations are fundamental to the generalization of trigonometric functions beyond acute angles and underpin transformations in advanced mathematical analysis.

Compound Angle Identities for Sine, Cosine, and Tangent

Let $A$ and $B$ be arbitrary real angles. The standard compound angle identities are as follows:

Identity: The sine of the sum and difference of two angles:

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

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

The cosine expressions for sum and difference:

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

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

The tangent sum and difference are given by:

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

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

Each of these identities is valid for all $A, B$ where the denominators are defined. For foundational trigonometric identities and further formal introduction, refer to Trigonometric Ratios Of Sine And Cosine and Trigonometric Ratios Of Tan, Cot, And Cosec.

Derivation of the Sine and Cosine Compound Angle Formulas

Consider two angles $A$ and $B$. Construct right triangles or employ Euler's formula to combine angles, then expand the product using trigonometric relations. For example, using the identities $\cos (X+Y) = \cos X \cos Y - \sin X \sin Y$ and $\sin (X+Y) = \sin X \cos Y + \cos X \sin Y$, one derives the sine and cosine formulas above, which constitute the backbone of compound angle transformations.

By extension, the tangent identity follows from the sine and cosine sums:

$\tan(A + B) = \dfrac{\sin(A + B)}{\cos(A + B)} = \dfrac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}$

Dividing numerator and denominator by $\cos A \cos B$ yields the tangent formula stated previously.

Compound Angle Identities Involving Products and Differences

The following results convert products of trigonometric functions into sums or differences:

$2 \sin A \cos B = \sin(A+B) + \sin(A-B)$

$2 \cos A \cos B = \cos(A+B) + \cos(A-B)$

$2 \sin A \sin B = \cos(A-B) - \cos(A+B)$

$2 \cos A \sin B = \sin(A+B) - \sin(A-B)$

Such transformations simplify expressions involving products and facilitate the integration and manipulation of trigonometric expressions. Their derivation follows from directly applying the compound angle identities by expressing one variable in terms of the other.

Trigonometric Ratios for Multiple and Submultiple Angles

Compounding the same angle enables the derivation of double, triple, and higher-angle identities. For example, the following hold for any angle $A$:

$\sin 2A = 2 \sin A \cos A$

$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$

$\tan 2A = \dfrac{2 \tan A}{1 - \tan^2 A}$

$\sin 3A = 3 \sin A - 4 \sin^3 A$

$\cos 3A = 4 \cos^3 A - 3 \cos A$

$\tan 3A = \dfrac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

These can be derived by repeated application of the compound angle identities as shown in trigonometric function expansions. For a rigorous study on multiples, see trigonometric functions of multiple angles.

Special Results and Inter-Relations of Compound Angle Expressions

Certain symmetric and difference expressions arise from the sum and difference formulas, for instance:

$\sin A \sin B = \dfrac{1}{2}[\cos(A-B) - \cos(A+B)]$

$\cos A \cos B = \dfrac{1}{2}[\cos(A+B) + \cos(A-B)]$

$\sin A \cos B = \dfrac{1}{2}[\sin(A+B) + \sin(A-B)]$

For if $A + B + C = \pi$, then $\cot A \cot B + \cot B \cot C + \cot C \cot A = 1$, deduced using cotangent addition formulas.

Exam-level questions may involve expressing higher-order trigonometric expressions using such relations and verifying identities under given constraints.

Typical Problem Patterns Using Compound Angle Identities

JEE Main and related examinations frequently require manipulation of compound angle relations to simplify expressions, calculate unknown angles, or convert between products and sums of trigonometric functions. For trigonometric ratios with specific argument values, reference is available in Inverse Trigonometric Functions and Trigonometric Ratios Of Compound Angles.

• Sum-to-product and product-to-sum transformations
• Solving equations involving compound arguments
• Establishing identities with $\sin(A+B)$ or $\tan(A-B)$
• Proof of constraints under angle sum properties
• Reduction of multiple angles to single functions

Mastery of these identities is necessary for efficiently addressing a range of trigonometric expressions and their geometric interpretations. Related geometric approaches may be further explored in Properties Of Triangle And Height And Distance.

Illustrative Problems Based on Compound Angle Ratios

Example: Evaluate $\sin 75^\circ$ using compound angle identity.

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

$= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

$= \dfrac{1}{\sqrt{2}}\cdot \dfrac{\sqrt{3}}{2} + \dfrac{1}{\sqrt{2}}\cdot \dfrac{1}{2}$

$= \dfrac{\sqrt{3} + 1}{2\sqrt{2}}$

Solution: All values substituted are from the standard trigonometric table.

Example: Prove that $\tan 3A = \dfrac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$.

Apply the tangent of sum: $\tan(2A + A) = \dfrac{\tan 2A + \tan A}{1 - \tan 2A \tan A}$

First, $\tan 2A = \dfrac{2 \tan A}{1 - \tan^2 A}$

So, $\tan 3A = \dfrac{\dfrac{2 \tan A}{1 - \tan^2 A} + \tan A}{1 - \dfrac{2 \tan A}{1 - \tan^2 A} \cdot \tan A}$

Simplify numerator:

$\dfrac{2 \tan A + \tan A (1 - \tan^2 A)}{1 - \tan^2 A}$

$= \dfrac{2 \tan A + \tan A - \tan^3 A}{1 - \tan^2 A}$

$= \dfrac{3 \tan A - \tan^3 A}{1 - \tan^2 A}$

Denominator:

$1 - \dfrac{2 \tan^2 A}{1 - \tan^2 A}$

$= \dfrac{1 - \tan^2 A - 2 \tan^2 A}{1 - \tan^2 A}$

$= \dfrac{1 - 3 \tan^2 A}{1 - \tan^2 A}$

Thus, $\tan 3A = \dfrac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

This method prevents algebraic miscalculations by isolating each transformation.

Common Calculation Errors in Compound Angle Problems

• Misapplication of sine and cosine sign conventions
• Miscalculation of tangent formulas due to denominator errors
• Confusing sum and difference in argument ordering
• Omitting denominator domain restrictions for tangent relations

To avoid such errors, clearly write each compound angle identity stepwise and confirm angle measures and quadrant signs as relevant. For foundational skills, revisit Understanding Trigonometry.