Signs and Allied Angles | JEE Main Trigonometry & Inverse Trigonometry Crash Course | Mathematicon

Signs and Allied Angles: Your JEE Main Trigonometry Toolkit

Welcome to a crucial lesson in your JEE Main trigonometry journey! Understanding the signs of trigonometric ratios in different quadrants and mastering allied angles will significantly boost your problem-solving speed and accuracy. This lesson equips you with the essential tools to tackle a wide range of trigonometric problems.

1. ASTC Rule: Signs of Trigonometric Ratios in Four Quadrants

Imagine the coordinate plane divided into four quadrants. The ASTC rule (All Silver Tea Cups) is a handy mnemonic to remember which trigonometric ratios are positive in each quadrant.

Quadrant I (0° - 90°): All trigonometric ratios ( $\sin$ , $\cos$ , $\tan$ , $\csc$ , $\sec$ , $\cot$ ) are positive.
Quadrant II (90° - 180°): Sine ( $\sin$ ) and its reciprocal, cosecant ( $\csc$ ), are positive.
Quadrant III (180° - 270°): Tangent ( $\tan$ ) and its reciprocal, cotangent ( $\cot$ ), are positive.
Quadrant IV (270° - 360°): Cosine ( $\cos$ ) and its reciprocal, secant ( $\sec$ ), are positive.

Intuition: Think about the unit circle. The sign of $\sin\theta$ is determined by the y-coordinate, and the sign of $\cos\theta$ is determined by the x-coordinate. $\tan\theta$ is simply $\frac{\sin\theta}{\cos\theta}$ , so its sign depends on the signs of sine and cosine.

For example, consider an angle of 150° (Quadrant II). Since sine is positive in Quadrant II, $\sin(150°)$ is positive. Cosine and tangent will be negative in this quadrant.

2. Allied Angles: Transforming Trigonometric Ratios

Allied angles are angles related to a given angle $\theta$ by multiples of 90° (i.e., $\frac{\pi}{2}$ radians). The most common allied angles are $(90° \pm \theta)$ , $(180° \pm \theta)$ , $(270° \pm \theta)$ , and $(360° \pm \theta)$ . Understanding how trigonometric ratios transform for these angles is essential.

3. Reduction Formulas for Allied Angles

Reduction formulas allow us to express trigonometric ratios of allied angles in terms of trigonometric ratios of $\theta$ . Here's the general principle:

Check the Quadrant: Determine which quadrant the allied angle lies in.
Determine the Sign: Based on the ASTC rule, determine the sign of the trigonometric ratio in that quadrant.
90°/270° Change, 180°/360° No Change:
- If the angle is a multiple of 90° (odd multiple of $\frac{\pi}{2}$ ), i.e., (90° ± θ) or (270° ± θ), the trigonometric ratio changes: $\sin \leftrightarrow \cos$ , $\tan \leftrightarrow \cot$ , $\sec \leftrightarrow \csc$ .
- If the angle is a multiple of 180° (even multiple of $\frac{\pi}{2}$ ), i.e., (180° ± θ) or (360° ± θ), the trigonometric ratio remains the same.

Important Formulas and Explanations

\sin(90° - \theta) = \cos\theta, \quad \cos(90° - \theta) = \sin\theta

Explanation: (90° - θ) lies in Quadrant I, where both sine and cosine are positive. Since we have 90°, sine changes to cosine, and cosine changes to sine.

Example: $\sin(90° - 30°) = \cos(30°) = \frac{\sqrt{3}}{2}$

\sin(180° - \theta) = \sin\theta, \quad \cos(180° - \theta) = -\cos\theta

Explanation: (180° - θ) lies in Quadrant II. Sine is positive in Quadrant II, so $\sin(180° - \theta)$ remains positive and sine. Cosine is negative in Quadrant II, so $\cos(180° - \theta)$ becomes $-\cos\theta$ . Since we have 180°, the trigonometric ratios don't change.

Example: $\cos(180° - 60°) = -\cos(60°) = -\frac{1}{2}$

\tan(180° + \theta) = \tan\theta

Explanation: (180° + θ) lies in Quadrant III. Tangent is positive in Quadrant III, so $\tan(180° + \theta)$ remains positive and tangent. Since we have 180°, the trigonometric ratio doesn't change.

Example: $\tan(180° + 45°) = \tan(45°) = 1$

4. Negative Angle Identities

\sin(-\theta) = -\sin\theta, \quad \cos(-\theta) = \cos\theta

Explanation: A negative angle $-\theta$ can be visualized as a clockwise rotation from the positive x-axis. The angle $-\theta$ lies in Quadrant IV. In Quadrant IV, cosine is positive, while sine is negative. Thus, $\cos(-\theta) = \cos\theta$ and $\sin(-\theta) = -\sin\theta$ . Tangent follows suit: $\tan(-\theta) = \frac{\sin(-\theta)}{\cos(-\theta)} = \frac{-\sin\theta}{\cos\theta} = -\tan\theta$ .

Example: $\sin(-30°) = -\sin(30°) = -\frac{1}{2}$

Tips for Solving Problems

Tip 1: Always visualize the angle on the coordinate plane to determine the quadrant. This helps in correctly applying the ASTC rule.

Tip 2: Break down complex angles into allied angles. For instance, $\sin(210°) = \sin(180° + 30°)$ .

Tip 3: Practice, practice, practice! The more problems you solve, the more comfortable you'll become with these transformations.

Common Mistakes to Avoid

Mistake 1: Forgetting to change the trigonometric ratio when using (90° ± θ) or (270° ± θ). Remember, sine becomes cosine, and vice versa.

Mistake 2: Incorrectly determining the sign of the trigonometric ratio in a particular quadrant. Double-check the ASTC rule!

Mistake 3: Confusing negative angle identities. Remember, only cosine "absorbs" the negative sign.

JEE-Specific Tricks

Trick 1: Many JEE problems involve multiple angle transformations. Be systematic and apply the rules step-by-step.

Trick 2: Look for opportunities to simplify expressions using allied angle identities before substituting values. This can save you time and reduce the chance of errors.

Trick 3: Remember that angles can be co-terminal. For example, 390° is the same as 30° (390° = 360° + 30°).

By mastering these concepts and practicing regularly, you'll be well-equipped to tackle even the most challenging problems involving signs and allied angles in your JEE Main preparation!