Compound angle formulas allow us to evaluate functions at non-standard angles such as 75^° or 15^° by splitting them into known angles like 45^° + 30^°. Note the sign reversal in the cosine expansion: a plus inside becomes a minus between the product terms.

Compound angle formulas allow us to evaluate functions at non-standard angles such as 75^° or 15^° by splitting them into known angles like 45^° + 30^°. Note the sign reversal in the cosine expansion: a plus inside becomes a minus between the product terms.

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Answer: Key Class 11 Trigonometry review complete: Master the radian conversion $\theta = l/r$ , the ASTC quadrant sign rules, compound angle expansions, double angle relations $\cos 2x$ and $\sin 2x$ , and sum-to-product CD formulas.

Step-by-step solution

Step 1: Master Radian Measure and Arc Length

In Class 11, angles are viewed as real numbers measured in radians rather than only degrees. An angle $\theta$ subtended by an arc of length $l$ in a circle of radius $r$ is given by $\theta = \frac{l}{r}$ . To convert degrees to radians, multiply by $\frac{\pi}{180}$ , and to convert radians to degrees, multiply by $\frac{180}{\pi}$ .

Step 2: Understand Signs of Functions via the ASTC Rule

The sign of trigonometric functions depends on which quadrant the terminal side of the angle lies in. We remember this using the rule 'All Silver Tea Cups' (ASTC). Furthermore, trigonometric functions have specific parity: $\cos(-x) = \cos(x)$ is an even function, while $\sin(-x) = -\sin(x)$ and $\tan(-x) = -\tan(x)$ are odd functions.

Step 3: Apply Compound Angle Addition Identities

Compound angle formulas allow us to evaluate functions at non-standard angles such as $75^\circ$ or $15^\circ$ by splitting them into known angles like $45^\circ + 30^\circ$ . Note the sign reversal in the cosine expansion: a plus inside becomes a minus between the product terms.

Step 4: Evaluate Double Angle and Half Angle Formulas

Double-angle formulas are derived directly from the addition formulas by setting $A = B = x$ . The cosine double angle has 3 useful forms, which are frequently used to express squared terms in linear forms: $\cos^2 x = \frac{1 + \cos(2x)}{2}$ and $\sin^2 x = \frac{1 - \cos(2x)}{2}$ .

Step 5: Apply Transformation Formulas (Product to Sum and Sum to Product)

Transformation identities convert sums of sines or cosines into products, which is essential for proving trigonometric identities and finding zeros. When dealing with $\cos C - \cos D$ , remember the minus sign: $\cos C - \cos D = -2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right) = 2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{D-C}{2}\right)$ .

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