What is the Sign of Trigonometric Ratios? | Simple Explanation for Class 10

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What is the Sign of Trigonometric Ratios in Different Quadrants?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The sign of trigonometric ratios (like sine, cosine, tangent) tells us if their value is positive (+) or negative (-) depending on which quadrant the angle falls in. This is important because angles can go beyond 90 degrees, and their ratios still have specific positive or negative signs.

Simple Example

Quick Example

Imagine your mobile data usage. If you use 1GB, it's a positive use. If you return 1GB unused from a pack, it's like a negative use (a saving). Similarly, for angles, if a 'sin' value is +0.5, it means one thing, but if it's -0.5, it means something else about its position in the coordinate system.

Worked Example

Step-by-Step

Let's find the sign of sin(150 degrees).
---1. First, identify the quadrant for 150 degrees. A full circle is 360 degrees.
---2. Quadrant I is 0-90 degrees. Quadrant II is 90-180 degrees. Quadrant III is 180-270 degrees. Quadrant IV is 270-360 degrees.
---3. Since 150 degrees is between 90 degrees and 180 degrees, it falls in Quadrant II.
---4. In Quadrant II, only sine (sin) and cosecant (cosec) ratios are positive. All other ratios (cosine, tangent, secant, cotangent) are negative.
---5. Since we are looking for sin(150 degrees) and sine is positive in Quadrant II, the sign will be positive.
---Answer: The sign of sin(150 degrees) is positive (+).

Why It Matters

Understanding these signs is super important in fields like Physics to calculate forces and trajectories, and in Engineering for designing structures. Even AI/ML algorithms use these concepts for pattern recognition and data analysis, helping engineers build smarter tech.

Common Mistakes

MISTAKE: Thinking all trigonometric ratios are always positive. | CORRECTION: Remember that only specific ratios are positive in specific quadrants. Use the 'All Silver Tea Cups' or 'Add Sugar To Coffee' rule to remember: All (Q1), Sine (Q2), Tan (Q3), Cos (Q4) are positive.

MISTAKE: Confusing the quadrants or their degree ranges. Forgetting that Quadrant I is 0-90, II is 90-180, etc. | CORRECTION: Always draw a simple coordinate plane and label the quadrants and their degree ranges (0, 90, 180, 270, 360) before solving.

MISTAKE: Applying the positive rule for sine to cosecant and vice versa, but forgetting about other reciprocal pairs (cos/sec, tan/cot). | CORRECTION: If a ratio (like sin) is positive in a quadrant, its reciprocal (cosec) will also be positive in that same quadrant. The same applies for negative signs.

Practice Questions

Try It Yourself

QUESTION: What is the sign of tan(210 degrees)? | ANSWER: Positive (+)

QUESTION: In which quadrant would cos(theta) be negative and sin(theta) be positive? | ANSWER: Quadrant II

QUESTION: If an angle 'A' is such that 270 < A < 360 degrees, what will be the sign of sec(A) and cot(A)? | ANSWER: sec(A) will be positive (+), cot(A) will be negative (-)

MCQ

Quick Quiz

Which of the following statements is TRUE?

In Quadrant III, only cosine is positive.

In Quadrant II, tangent is positive.

In Quadrant IV, sine is positive.

In Quadrant I, all trigonometric ratios are positive.

The Correct Answer Is:

In Quadrant I (0-90 degrees), both x and y coordinates are positive, so all ratios (which depend on x, y, and r) are positive. Options A, B, and C are incorrect based on the quadrant sign rules.

Real World Connection

In the Real World

When ISRO scientists launch satellites, they constantly calculate angles and trajectories. Understanding the signs of trigonometric ratios helps them precisely determine the satellite's position and direction in space, ensuring it reaches its correct orbit around Earth.

Key Vocabulary

Key Terms

QUADRANT: One of the four regions into which a coordinate plane is divided by the x and y axes. | TRIGONOMETRIC RATIO: Ratios of sides of a right-angled triangle, like sine, cosine, tangent. | ANGLE: The amount of turn between two lines or surfaces meeting at a common point. | COORDINATE PLANE: A 2D surface defined by two perpendicular lines (x-axis and y-axis) used to locate points.

What's Next

What to Learn Next

Great job understanding the signs! Next, you should explore 'Trigonometric Identities' and 'Reduction Formulas'. These build directly on knowing quadrant signs and will help you simplify complex trigonometric expressions, which is super useful for higher-level math and science.