Trigonometry: Easy Trick to Find Values of 180°, 270° and 360° Angles
Do you get confused when a Trigonometry question contains angles like 120°, 150°, 210°, 225°, 270°, 300° or 330°?
Many students think that they have to memorize a new Trigonometry table for these angles. You do not.
There is a simple method called the Reference Angle Method. Once you understand this method, you can find the values of many Trigonometric angles using the same basic table of 0°, 30°, 45°, 60° and 90°.
First find the small familiar angle.
Then find the sign using the quadrant.
Table of Contents
- The Basic Idea
- What is a Reference Angle?
- Understand the Four Quadrants
- Which Trigonometric Ratio is Positive?
- The 180° Trick
- The 270° Trick
- The 360° Trick
- Examples from 90° to 180°
- Examples from 180° to 270°
- Examples from 270° to 360°
- Angles Greater Than 360°
- Negative Angles
- The Fastest Exam Method
- Common Mistakes
- Practice Questions
- Answers
- FAQs
1. The Basic Idea
Before learning this trick, you only need to know the standard values of these five angles:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Not Defined |
These are enough for a large number of questions.
Remember one important point:
They are exactly the same value. In this article, we will consistently use 1/√2.
2. What is a Reference Angle?
A reference angle is the small positive angle between the terminal side of an angle and the x-axis.
The important idea is very simple:
For example:
150° is close to 180°.
180° − 150° = 30°
So the reference angle is 30°.
Therefore, the basic numerical value will come from the 30° table. Since 150° lies in the second quadrant, we then apply the correct sign.
3. Understand the Four Quadrants
A complete circle is divided into four parts called quadrants.
| Quadrant | Angle Range |
|---|---|
| I | 0° to 90° |
| II | 90° to 180° |
| III | 180° to 270° |
| IV | 270° to 360° |
Easy picture to remember
0° → 90° → 180° → 270° → 360°
Think of moving around a circle in the anticlockwise direction.
4. Which Trigonometric Ratio is Positive?
This is the most important part of the trick.
| Quadrant | Positive Ratios |
|---|---|
| I | All |
| II | sin, cosec |
| III | tan, cot |
| IV | cos, sec |
Memory trick: ASTC
You may remember the four quadrants using:
It means:
- All are positive in Quadrant I.
- Sine is positive in Quadrant II.
- Tangent is positive in Quadrant III.
- Cosine is positive in Quadrant IV.
Their reciprocal ratios have the same sign: cosec follows sin, cot follows tan, and sec follows cos.
5. The 180° Trick
When an angle is between 90° and 180°, subtract it from 180°.
Example 1: sin 150°
Step 1:
180° − 150° = 30°
Step 2:
sin 30° = 1/2
Step 3:
150° is in Quadrant II, where sine is positive.
Therefore:
sin 150° = 1/2
Example 2: cos 150°
Reference angle = 180° − 150° = 30°
cos 30° = √3/2
But cosine is negative in Quadrant II.
cos 150° = −√3/2
Example 3: tan 150°
Reference angle = 30°
tan 30° = 1/√3
Tangent is negative in Quadrant II.
tan 150° = −1/√3
6. The 270° Trick
Angles between 180° and 270° are in Quadrant III.
For an angle in Quadrant III:
Example: sin 210°
210° − 180° = 30°
sin 30° = 1/2
Sine is negative in Quadrant III.
sin 210° = −1/2
Example: cos 225°
225° − 180° = 45°
cos 45° = 1/√2
Cosine is negative in Quadrant III.
cos 225° = −1/√2
Example: tan 240°
240° − 180° = 60°
tan 60° = √3
Tangent is positive in Quadrant III.
tan 240° = √3
7. The 360° Trick
Angles between 270° and 360° are in Quadrant IV.
In this quadrant:
Example: sin 330°
360° − 330° = 30°
sin 30° = 1/2
Sine is negative in Quadrant IV.
sin 330° = −1/2
Example: cos 300°
360° − 300° = 60°
cos 60° = 1/2
Cosine is positive in Quadrant IV.
cos 300° = 1/2
Example: tan 315°
360° − 315° = 45°
tan 45° = 1
Tangent is negative in Quadrant IV.
tan 315° = −1
8. Examples from 90° to 180°
Remember:
Example 1: sin 120°
180° − 120° = 60°
sin 60° = √3/2
Sine is positive in Quadrant II.
Answer: sin 120° = √3/2
Example 2: cos 135°
180° − 135° = 45°
cos 45° = 1/√2
Cosine is negative in Quadrant II.
Answer: cos 135° = −1/√2
Example 3: tan 120°
180° − 120° = 60°
tan 60° = √3
Tangent is negative in Quadrant II.
Answer: tan 120° = −√3
Example 4: sin 135°
Reference angle = 180° − 135° = 45°
sin 45° = 1/√2
Sine is positive in Quadrant II.
Answer: sin 135° = 1/√2
Example 5: cos 150°
Reference angle = 30°
cos 30° = √3/2
Cosine is negative.
Answer: cos 150° = −√3/2
9. Examples from 180° to 270°
Here the trick is:
Example 1: sin 225°
225° − 180° = 45°
sin 45° = 1/√2
Sine is negative in Quadrant III.
Answer: sin 225° = −1/√2
Example 2: cos 210°
210° − 180° = 30°
cos 30° = √3/2
Cosine is negative in Quadrant III.
Answer: cos 210° = −√3/2
Example 3: tan 225°
225° − 180° = 45°
tan 45° = 1
Tangent is positive in Quadrant III.
Answer: tan 225° = 1
Example 4: sin 240°
240° − 180° = 60°
sin 60° = √3/2
Sine is negative.
Answer: sin 240° = −√3/2
Example 5: cos 255°
255° − 180° = 75°
Here the reference angle is 75°, which is not one of the standard 30°, 45° or 60° angles.
So the reference-angle method still tells us the correct relationship and sign, but we cannot obtain a simple exact value from the basic standard-value table alone.
This is an important point: the trick does not magically create a standard value for every angle. It helps us reduce familiar angles and determine signs.
10. Examples from 270° to 360°
Here:
Example 1: sin 300°
360° − 300° = 60°
sin 60° = √3/2
Sine is negative in Quadrant IV.
Answer: sin 300° = −√3/2
Example 2: cos 330°
360° − 330° = 30°
cos 30° = √3/2
Cosine is positive in Quadrant IV.
Answer: cos 330° = √3/2
Example 3: tan 300°
360° − 300° = 60°
tan 60° = √3
Tangent is negative in Quadrant IV.
Answer: tan 300° = −√3
Example 4: sin 315°
360° − 315° = 45°
sin 45° = 1/√2
Sine is negative in Quadrant IV.
Answer: sin 315° = −1/√2
Example 5: cos 315°
Reference angle = 45°
cos 45° = 1/√2
Cosine is positive in Quadrant IV.
Answer: cos 315° = 1/√2
11. What About Angles Greater Than 360°?
This is another useful trick.
One complete revolution is:
Therefore, for an angle greater than 360°, subtract 360° repeatedly until you get an angle between 0° and 360°.
Example: sin 390°
390° − 360° = 30°
Therefore:
sin 390° = sin 30° = 1/2
Example: cos 420°
420° − 360° = 60°
Therefore:
cos 420° = cos 60° = 1/2
Example: tan 765°
Subtract 720°:
765° − 720° = 45°
Therefore:
tan 765° = tan 45° = 1
12. What About Negative Angles?
Negative angles can also be handled without memorizing a new table.
Important relationships are:
- sin(−θ) = −sin θ
- cos(−θ) = cos θ
- tan(−θ) = −tan θ
Example: sin(−30°)
sin 30° = 1/2
Therefore:
sin(−30°) = −1/2
Example: cos(−60°)
cos(−60°) = cos 60°
Answer = 1/2
Example: tan(−45°)
tan(−45°) = −tan 45°
Answer = −1
13. The Fastest Exam Method
When you see an unfamiliar angle, do not panic.
Follow these 3 steps:
STEP 1: Find the quadrant.
STEP 2: Find the reference angle.
STEP 3: Take the standard value and apply the correct sign.
Example: Find sin 330°
Step 1: 330° lies in Quadrant IV.
Step 2: Reference angle = 360° − 330° = 30°.
Step 3: sin 30° = 1/2, but sine is negative in Quadrant IV.
Therefore, sin 330° = −1/2
14. One-Minute Master Chart
| Angle Range | Reference Angle | Positive Ratio |
|---|---|---|
| 0°–90° | Same angle | All |
| 90°–180° | 180° − θ | sin |
| 180°–270° | θ − 180° | tan |
| 270°–360° | 360° − θ | cos |
Remember:
QII → Sin
QIII → Tan
QIV → Cos
15. Common Mistakes Weak Students Make
Mistake 1: Forgetting the sign
Students often find the correct reference angle but forget whether the answer should be positive or negative.
Solution: Always find the quadrant first.
Mistake 2: Using 180° − θ everywhere
This works for Quadrant II, but not for every quadrant.
Use the correct formula:
- Quadrant II → 180° − θ
- Quadrant III → θ − 180°
- Quadrant IV → 360° − θ
Mistake 3: Thinking 150° has the same sign as 30°
The numerical value comes from 30°, but the sign depends on the quadrant.
For example:
sin 150° = +1/2
But:
cos 150° = −√3/2
Mistake 4: Thinking every angle has a simple standard value
Angles such as 75°, 20° or 17° are not part of the basic standard-value table.
Reference angles help with the relationship and sign, but an exact simple value may require additional formulas or information.
16. Practice Questions
Try these without looking at the answers.
- Find sin 120°.
- Find cos 135°.
- Find tan 150°.
- Find sin 210°.
- Find cos 225°.
- Find tan 240°.
- Find sin 300°.
- Find cos 315°.
- Find tan 330°.
- Find sin 390°.
- Find cos 420°.
- Find tan 765°.
- Find sin(−30°).
- Find cos(−60°).
- Find tan(−45°).
17. Answers
- sin 120° = √3/2
- cos 135° = −1/√2
- tan 150° = −1/√3
- sin 210° = −1/2
- cos 225° = −1/√2
- tan 240° = √3
- sin 300° = −√3/2
- cos 315° = 1/√2
- tan 330° = −1/√3
- sin 390° = 1/2
- cos 420° = 1/2
- tan 765° = 1
- sin(−30°) = −1/2
- cos(−60°) = 1/2
- tan(−45°) = −1
18. Frequently Asked Questions
Do I need to memorize a separate table for 120°, 150°, 210°, 225°, 300° and 330°?
No. You can find these values using the reference angle method and the standard values of 0°, 30°, 45°, 60° and 90°.
What is the easiest way to remember the signs?
Remember ASTC: All, Sin, Tan, Cos.
How do I find the reference angle of 150°?
Since 150° is in Quadrant II: 180° − 150° = 30°.
How do I find the reference angle of 225°?
Since 225° is in Quadrant III: 225° − 180° = 45°.
How do I find the reference angle of 315°?
Since 315° is in Quadrant IV: 360° − 315° = 45°.
What should I do with an angle greater than 360°?
Subtract 360° as many times as necessary until the angle falls between 0° and 360°. Then use the quadrant and reference-angle method.
19. Final 30-Second Revision
Step 1: Find the quadrant.
Step 2: Find the reference angle.
QII → 180° − θ
QIII → θ − 180°
QIV → 360° − θ
Step 3: Use the standard value.
Step 4: Apply the correct sign.
Big Angle → Small Angle → Standard Value → Correct Sign
That is the whole trick.
You do not need to memorize a separate table for every angle. Once the basic 0°, 30°, 45°, 60° and 90° values are understood, the reference-angle method can handle many other angles.
Related Topic: First learn the basic Trigonometry table, then use this reference-angle trick to handle angles beyond 90°.
Read More
- Basic Concept of Trigonometry: Measurement of Angles
- Angles, Degrees, and Radians Explained
- Different Systems of Measurement of Angles in Trigonometry: Sexagesimal System, Centesimal System and Circular System
- Introduction of Basic Trigonometry Formulas
- All Basic Trigonometric Identities Formulas
- Geometry Behind Trigonometric Identities
- Difference between Trigonometric Identity (trig identity) and Trigonometric Equation (trig equation)
- Learn All Trigonometric Ratios in 30 Seconds
- Mastering All Six Trigonometric Ratios in a Fun and Easy Way: Learn Sine, Cosine, and Tangent Ratios Explained
- Practice Questions with Answers to Learn All Six Trigonometric Ratios
- Why sin²θ + cos²θ = 1 Always Holds True?
- Where Does sin²θ + cos²θ = 1 Come From?
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