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What is the Sign Convention for Trigonometric Ratios (basic)?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The sign convention for trigonometric ratios tells us whether sine, cosine, tangent, and their reciprocals are positive or negative in different parts of a coordinate plane. It helps us understand how these ratios change based on the angle's position.
Simple Example
Quick Example
Imagine you're walking on a cricket field. If you walk towards the pavilion (positive X-axis), your 'x' coordinate is positive. If you walk towards the scoreboard (positive Y-axis), your 'y' coordinate is positive. Similarly, trigonometric ratios have positive or negative 'signs' depending on which quadrant your angle lands in, like deciding if a run is positive (scored) or negative (given away).
Worked Example
Step-by-Step
PROBLEM: Determine the sign of sin(150 degrees).
STEP 1: Identify the quadrant for 150 degrees. Since 90 degrees < 150 degrees < 180 degrees, the angle 150 degrees lies in the Second Quadrant.
---STEP 2: Recall the sign convention for the Second Quadrant. In the Second Quadrant, only sine (sin) and its reciprocal, cosecant (cosec), are positive.
---STEP 3: Since 150 degrees is in the Second Quadrant and sine is positive in this quadrant, sin(150 degrees) will be positive.
---ANSWER: The sign of sin(150 degrees) is positive.
Why It Matters
Understanding sign conventions is crucial for engineers designing bridges, physicists tracking satellite orbits, and even AI developers building navigation systems. It helps them accurately calculate directions, forces, and positions, ensuring everything from safe flights to precise robotic movements.
Common Mistakes
MISTAKE: Thinking all trigonometric ratios are positive in every quadrant. | CORRECTION: Remember the 'All Silver Tea Cups' (ASTC) rule or 'Add Sugar To Coffee' rule to correctly identify which ratios are positive in each quadrant.
MISTAKE: Confusing the signs of sine and cosine in the Third Quadrant. | CORRECTION: In the Third Quadrant, both x and y coordinates are negative. Since sin depends on y and cos depends on x, both sin and cos are negative here.
MISTAKE: Forgetting that reciprocal ratios (like cosec for sin) have the same sign as their main ratio. | CORRECTION: If sin is positive, cosec is positive. If tan is negative, cot is negative. Their signs always match.
Practice Questions
Try It Yourself
QUESTION: What is the sign of cos(210 degrees)? | ANSWER: Negative
QUESTION: An angle 'theta' is in the Fourth Quadrant. What are the signs of tan(theta) and sec(theta)? | ANSWER: tan(theta) is negative, sec(theta) is positive.
QUESTION: If sin(A) is positive and cos(A) is negative, in which quadrant does angle A lie? | ANSWER: Second Quadrant
MCQ
Quick Quiz
In which quadrant is only tan (and cot) positive?
First Quadrant
Second Quadrant
Third Quadrant
Fourth Quadrant
The Correct Answer Is:
C
In the First Quadrant, all ratios are positive. In the Second, only sin is positive. In the Third, both x and y are negative, making tan (y/x) positive. In the Fourth, only cos is positive.
Real World Connection
In the Real World
When ISRO scientists launch rockets, they use trigonometry to calculate trajectories. The sign convention helps them determine if a rocket's angle of ascent or descent is in the correct 'direction' relative to the Earth, ensuring precise navigation and successful missions, like placing satellites into orbit.
Key Vocabulary
Key Terms
QUADRANT: One of the four regions into which a coordinate plane is divided by the x and y axes. | SINE: A trigonometric ratio relating the opposite side to the hypotenuse in a right-angled triangle. | COSINE: A trigonometric ratio relating the adjacent side to the hypotenuse. | TANGENT: A trigonometric ratio relating the opposite side to the adjacent side.
What's Next
What to Learn Next
Great job understanding the signs! Next, explore 'Trigonometric Ratios of Specific Angles' like 30, 45, 60, and 90 degrees. This will help you find actual numerical values for these ratios, building on your knowledge of their signs.
