How to Find the Sign of Trigonometric Functions Using the Unit Circle Rule
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 in Different Quadrants
1. What is the sign of trigonometric functions?
The sign of trigonometric functions tells whether functions like sin, cos, and tan are positive or negative in different quadrants of the coordinate plane. In the Cartesian plane:
- Quadrant I: All trigonometric functions are positive.
- Quadrant II: Only sin θ and cosec θ are positive.
- Quadrant III: Only tan θ and cot θ are positive.
- Quadrant IV: Only cos θ and sec θ are positive.
2. How do you determine the sign of sin, cos, and tan in each quadrant?
You determine the sign of sin, cos, and tan using the ASTC rule based on the quadrant of the angle. The rules are:
- sin θ is positive in Quadrants I and II.
- cos θ is positive in Quadrants I and IV.
- tan θ is positive in Quadrants I and III.
3. Why is sine positive in the second quadrant?
Sine is positive in the second quadrant because the y-coordinate of any point in Quadrant II is positive. Since sin θ = y/r and the radius r is always positive, the sign of sin θ depends on y. In Quadrant II:
- x is negative
- y is positive
4. What is the ASTC rule in trigonometry?
The ASTC rule is a mnemonic used to remember the sign of trigonometric functions in each quadrant. It stands for:
- A (All) – All functions positive in Quadrant I
- S (Sine) – Only sine positive in Quadrant II
- T (Tangent) – Only tangent positive in Quadrant III
- C (Cosine) – Only cosine positive in Quadrant IV
5. What is the sign of trigonometric functions in Quadrant III?
In Quadrant III, only tan θ and cot θ are positive, while sine and cosine are negative. This is because:
- x is negative
- y is negative
- tan θ = y/x, so negative ÷ negative = positive
6. How do you find the sign of trigonometric ratios for an angle greater than 90°?
To find the sign of trigonometric ratios for angles greater than 90°, first identify the quadrant in which the angle lies. For example:
- 120° lies in Quadrant II, so sin 120° is positive and cos 120° is negative.
- 210° lies in Quadrant III, so tan 210° is positive.
7. What is the sign of trigonometric functions on the coordinate axes?
On the coordinate axes, some trigonometric functions are zero while others are positive or negative depending on direction. For example:
- At 0°: sin 0° = 0, cos 0° = 1
- At 90°: sin 90° = 1, cos 90° = 0
- At 180°: sin 180° = 0, cos 180° = -1
8. How do you use the unit circle to determine the sign of trigonometric functions?
You use the unit circle by identifying the coordinates (x, y) of the point corresponding to the angle. In the unit circle:
- cos θ = x
- sin θ = y
- tan θ = y/x
9. What is the sign of sec, cosec, and cot in different quadrants?
The signs of sec, cosec, and cot depend on their reciprocal relationships. Since:
- sec θ = 1/cos θ
- cosec θ = 1/sin θ
- cot θ = 1/tan θ
- In Quadrant II, cosec θ is positive
- In Quadrant III, cot θ is positive
- In Quadrant IV, sec θ is positive
10. Can you give an example of finding the sign of a trigonometric function?
Yes, to find the sign of a trigonometric function, first identify the quadrant of the angle and apply the sign rules. Example:
- Find the sign of cos 150°.
- 150° lies in Quadrant II.
- In Quadrant II, cosine is negative.
