Compound Angle Formulas (Sum and Difference)

Compound Angle Formulas (Sum and Difference)

Welcome, future JEE champions! This lesson is your gateway to mastering compound angle formulas, a crucial tool in trigonometry. These formulas allow you to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. Mastering these formulas will significantly boost your problem-solving speed and accuracy in JEE Main.

Understanding Compound Angles

Imagine you know the trigonometric ratios of two angles, say $A$ and $B$. Can you find the trigonometric ratios of $A + B$ or $A - B$? That's exactly what compound angle formulas help us do! They provide a direct relationship between trigonometric functions of individual angles and their sums or differences.

1. Sine of Sum and Difference: $\sin(A \pm B)$

Let's start with the sine function. The formulas for $\sin(A + B)$ and $\sin(A - B)$ are:

Intuition: Notice the pattern. For $\sin(A + B)$, we have a "sine-cosine plus cosine-sine" structure. For $\sin(A - B)$, it's "sine-cosine minus cosine-sine". The sign between the terms matches the sign between $A$ and $B$ in the function's argument.

Derivation (Geometric Approach): While a rigorous proof requires geometry, let's build some intuition. Consider a unit circle. The $y$-coordinate of a point on the circle represents the sine of the angle. The formula essentially decomposes the $y$-coordinate of the angle $A+B$ into components related to $A$ and $B$ individually using projection.

Example: Suppose you need to find $\sin 75^\circ$. You can express $75^\circ$ as $45^\circ + 30^\circ$. Since you know the values of $\sin$ and $\cos$ for $45^\circ$ and $30^\circ$, you can directly apply the formula: $$\sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}}$$

2. Cosine of Sum and Difference: $\cos(A \pm B)$

Now, let's explore the cosine function. The formulas are:

Intuition: For cosine, we have "cosine-cosine minus sine-sine" for $\cos(A + B)$, and "cosine-cosine plus sine-sine" for $\cos(A - B)$. Notice the sign difference – a '+' sign in the argument corresponds to a '-' sign between the terms, and vice-versa.

Derivation (Geometric Approach): Similar to sine, the cosine formulas relate to the $x$-coordinate of a point on the unit circle, which is decomposed into components using projections related to angles $A$ and $B$.

Example: Let's calculate $\cos 15^\circ$. We can write $15^\circ$ as $45^\circ - 30^\circ$. Then, $$\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}}$$

3. Tangent of Sum and Difference: $\tan(A \pm B)$

The tangent formulas are derived from the sine and cosine formulas:

Derivation: We know that $\tan x = \frac{\sin x}{\cos x}$. So, $\tan(A + B) = \frac{\sin(A + B)}{\cos(A + B)}$. Substituting the sine and cosine formulas and dividing both numerator and denominator by $\cos A \cos B$, we get the tangent formula.

Example: Calculate $\tan 105^\circ$. We can express $105^\circ$ as $60^\circ + 45^\circ$. $$\tan 105^\circ = \tan(60^\circ + 45^\circ) = \frac{\tan 60^\circ + \tan 45^\circ}{1 - \tan 60^\circ \tan 45^\circ} = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}$$ Rationalizing, we get $\tan 105^\circ = -2 - \sqrt{3}$.

4. Cotangent of Sum and Difference: $\cot(A \pm B)$

The cotangent formulas are:

While not as commonly used, it's good to know:

$$cot(A + B) = \frac{cotAcotB - 1}{cotA + cotB}$$

$$cot(A - B) = \frac{cotAcotB + 1}{cotB - cotA}$$

These formulas can be derived using a similar approach to tangent, using the identity $cot x = \frac{cos x}{sin x}$.