Quick ASTC Rule Chart: Signs of Trig Functions Simplified
The concept of sign of trigonometric functions is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Knowing whether sine, cosine, tangent, and their reciprocals are positive or negative in each quadrant prevents mistakes and boosts confidence during calculations.
Understanding Sign of Trigonometric Functions
The sign of trigonometric functions tells us if sine, cosine, tangent, cosecant, secant, and cotangent are positive or negative for angles in different quadrants of the coordinate system. This concept is widely used in trigonometric ratios, solving trigonometric equations, and understanding trigonometric functions.
Signs in Each Quadrant (ASTC Rule)
To quickly remember signs of trigonometric functions in quadrants, we follow the ASTC rule (All Students Take Calculus):
2. **Quadrant II (90° to 180°):** Only sine and cosecant are positive. Others are negative.
3. **Quadrant III (180° to 270°):** Only tangent and cotangent are positive. Others are negative.
4. **Quadrant IV (270° to 360°):** Only cosine and secant are positive. Others are negative.
This quadrant-wise approach is crucial for board exams and competitive tests.
Here’s a helpful table to understand the sign of trigonometric functions in all four quadrants more clearly:
Sign of Trigonometric Functions Table
| Function | Quadrant I (0°–90°) |
Quadrant II (90°–180°) |
Quadrant III (180°–270°) |
Quadrant IV (270°–360°) |
|---|---|---|---|---|
| sin θ | + | + | – | – |
| cos θ | + | – | – | + |
| tan θ | + | – | + | – |
| cosec θ | + | + | – | – |
| sec θ | + | – | – | + |
| cot θ | + | – | + | – |
This table shows how quadrant position directly determines the positive or negative sign of trigonometric functions.
Worked Example – Solving a Problem
Let’s see a step-by-step example using the quadrant sign rules:
2. We use the identity: \( \csc^2 x - \cot^2 x = 1 \).
3. Substitute \(\cot x\):
4. Therefore, \( \csc x = \pm \dfrac{13}{12} \).
5. In quadrant IV, sine and cosec are negative. So, the answer is:
Final Answer: \( \csc x = -\dfrac{13}{12} \)
Tips, Tricks & Mnemonics
- ASTC Memory Rule: “All Students Take Calculus” — use the first letter of each word for each quadrant, starting from I to IV. All (all positive), Students (Sine), Take (Tangent), Calculus (Cosine).
- Check the reference angle and its quadrant before assigning the sign of any function.
- Remember: Positive only means non-negative, not maximum value.
Why the Sign of Trigonometric Functions Matters in Class 11 & 12
Students in class 11 and 12 regularly need to use signs of trigonometric functions in each quadrant when solving problems in trigonometric ratios, identities, equations and advanced concepts. This topic is vital for CBSE, JEE, NEET and other competitive exams.
Quick Revision Table
| Quadrant | Functions Positive (+) |
|---|---|
| I | All |
| II | Sine, Cosec |
| III | Tan, Cot |
| IV | Cos, Sec |
Common Mistakes to Avoid
- Forgetting which quadrant is being used when assigning a sign.
- Assuming the sign stays the same for co-functions like sin and cosec, or cos and sec, without confirmation from the quadrant chart.
- Not converting negative or large angles to their correct reference angle/quadrant.
Frequently Asked Questions
Q1: What are the signs of trigonometric functions in each quadrant?
A: In I: all positive; II: sine/cosec positive; III: tan/cot positive; IV: cos/sec positive.
Q2: Is sine positive in the 2nd quadrant?
A: Yes, both sine and cosec are positive in the second quadrant.
Q3: How can I know if a trig function is positive or negative for an angle like 210°?
A: First, find the quadrant (210° is in quadrant III), then use the table: tan and cot are positive, others negative.
Practice Problems
- Find the sign of tan(120°) and sec(330°).
- For angle –45°, which trigonometric functions are positive?
- Is cos(400°) positive or negative?
- List all trigonometric functions that are negative in the third quadrant.
Summary
We explored the idea of sign of trigonometric functions, their quadrant-wise distribution, solved step-by-step examples, and shared memory tricks. Regular revision and table practice with Vedantu makes these concepts easy and applicable in real problems. For more details, check our Trigonometric Functions and Trigonometry Table pages.
Related Topics and Further Study
FAQs on Sign of Trigonometric Functions: Master Quadrant Rules Fast
1. What are the signs of trigonometric functions in each quadrant?
The signs of the six main trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—vary by quadrant as follows: In Quadrant I, all functions are positive. In Quadrant II, only sine and cosecant are positive. In Quadrant III, only tangent and cotangent are positive. In Quadrant IV, only cosine and secant are positive. Understanding this is essential for solving problems involving angle values and their signs.
2. Is sine positive in the 2nd quadrant?
Yes, sine (sin θ) is positive in the 2nd quadrant. According to the ASTC (All Students Take Calculus) rule, only sine and its reciprocal cosecant are positive in Quadrant II, while cosine, tangent, and their reciprocals are negative.
3. How do I know if a trig function is positive or negative?
To determine if a trigonometric function is positive or negative, follow these steps:
1. Identify the quadrant in which the angle lies.
2. Apply the ASTC rule which states: Quadrant I – all positive, Quadrant II – sine positive, Quadrant III – tangent positive, Quadrant IV – cosine positive.
3. Check if the function matches one of those positive in the quadrant. If yes, it's positive; otherwise, negative.
Using the unit circle and memorizing the quadrant signs improves accuracy.
4. What is the ASTC rule in trigonometry?
The ASTC rule is a mnemonic to remember the sign of trigonometric functions in the four quadrants:
- A: All functions are positive in Quadrant I.
- S: Sine and cosecant are positive in Quadrant II.
- T: Tangent and cotangent are positive in Quadrant III.
- C: Cosine and secant are positive in Quadrant IV.
This helps students quickly identify the sign without confusion during exams.
5. How do signs of trigonometric functions apply in board exams?
In board exams, correctly identifying the sign of trigonometric functions is crucial for accurate answers, especially in solving equations and verifying identities. Errors in sign determination often lead to incorrect solutions. Familiarity with quadrant signs, practicing typical problems, and using visual aids like the ASTC chart can significantly reduce mistakes.
6. Why do so many students mix up sine and cosine signs in exams?
Students commonly mix up sine and cosine signs due to similar appearance of problems and misunderstanding of quadrant positioning. Since sine is positive in Quadrant II and cosine is negative there (and vice versa in Quadrant IV), confusion arises without clear mental mapping. Using the ASTC mnemonic and repeatedly practicing quadrant sign charts helps build confidence and accuracy.
7. Why is quadrant numbering important for function signs?
Quadrant numbering (I to IV) establishes the reference for determining which trigonometric functions are positive or negative. Each quadrant corresponds to a 90° sector in the unit circle, and the sign of functions depends on the x- and y-coordinates within those sectors. Correct numbering prevents sign errors and aids in interpreting angle measures beyond 0°–90°, including negative and >360° angles.
8. How to quickly recall the sign for 420° or –60°?
To recall signs for angles like 420° or –60°:
1. Normalize the angle by subtracting or adding 360° until it lies between 0° and 360°.
—For 420°, 420° – 360° = 60° (Quadrant I).
—For –60°, –60° + 360° = 300° (Quadrant IV).
2. Apply the ASTC rule based on the normalized quadrant.
This method simplifies sign determination for angles outside the standard 0°–360° range.
9. What is a common pitfall when using calculators for trig signs?
A common pitfall when using calculators is forgetting that calculators typically return principal angle values between 0° and 90°, without indicating the quadrant. Students may misinterpret the sign of the trigonometric value without adjusting for the correct quadrant, leading to incorrect answers. Always verify the quadrant of the input angle and manually apply sign rules accordingly.
10. Why do JEE and NEET often give negative answer options in such problems?
Competitive exams like JEE and NEET include negative answer options to test students’ understanding of sign rules in trigonometry. Many problems involve angles in different quadrants where functions can be negative. Including negative options challenges students to carefully apply quadrant sign rules (ASTC) rather than blindly computing magnitudes.
11. What are the benefits of using a signs of trigonometric functions calculator?
A signs of trigonometric functions calculator helps in:
- Quickly determining the positive or negative nature of trig functions for a given angle.
- Supporting revision and doubt-clearance during exam preparation.
- Visualizing quadrant-based sign changes.
While helpful, students should use calculators alongside conceptual understanding to avoid over-dependence.
12. How do the signs affect solving trigonometric equations?
Signs affect the solution of trigonometric equations because the value of a function changes with quadrant. Knowing the correct sign helps identify the correct solution interval and verify if the angle satisfies the equation. Ignoring signs can lead to missing valid solutions or choosing incorrect ones, especially in problems involving multiple quadrants and periodicity.
