What is the value of tan 15° and how is it calculated?
The concept of value of tan 15 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. This value is frequently asked in board exams, NTSE, Olympiads, JEE, and in trigonometry MCQs. Knowing the exact value and trick to calculate tan 15 can help you solve a variety of geometry and height-and-distance problems efficiently.
What Is Value of Tan 15?
The value of tan 15 refers to the trigonometric ratio of the angle 15 degrees, defined as the ratio of sine to cosine of 15°, or the ratio of the perpendicular to the base in a right-angled triangle with a 15° angle. You’ll find this concept applied in areas such as geometry, angle subtraction formulas, and competitive exam problem solving.
Key Formula for Value of Tan 15
Here’s the standard formula: \( \tan(15^\circ) = \tan(45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1+\tan 45^\circ \tan 30^\circ} \)
Cross-Disciplinary Usage
Value of tan 15 is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions—especially when dealing with angles, vectors, forces, and coordinate geometry.
Step-by-Step Illustration
- Express 15° as a difference: \( 15^\circ = 45^\circ - 30^\circ \)
- Recall standard trigonometric values: \( \tan 45^\circ = 1 \), \( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
- Apply the angle subtraction formula:\( \tan(45^\circ-30^\circ) = \frac{1-\frac{1}{\sqrt{3}}}{1+1 \times \frac{1}{\sqrt{3}}} \)
- Simplify numerator and denominator:Numerator: \( 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3}-1}{\sqrt{3}} \)
Denominator: \( 1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3}+1}{\sqrt{3}} \) - Combine fractions:\( \tan 15^\circ = \frac{\sqrt{3}-1}{\sqrt{3}+1} \)
- Rationalize denominator:Multiply numerator and denominator by \( \sqrt{3}-1 \):
\( \frac{(\sqrt{3}-1)^2}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{3-2\sqrt{3} +1}{2} = 2-\sqrt{3} \) - Final answer: \( \tan 15^\circ = 2 - \sqrt{3} \) ≈ 0.2679
Value Table (Fractions & Roots)
| Angle | Root/Fraction Form | Decimal Value |
|---|---|---|
| tan 15° | (√3 - 1)/(√3 + 1) or 2 − √3 |
0.2679 |
| cot 15° | (√3 + 1)/(√3 - 1) or 2 + √3 |
3.732 |
| tan 45° | 1 | 1.000 |
| tan 30° | 1/√3 | 0.577 |
| tan 75° | 2 + √3 | 3.732 |
Instant Answer: tan 15° = 2 − √3 ≈ 0.2679
Speed Trick or Vedic Shortcut
Here’s a quick memory trick: Whenever you see tan 15° in MCQs, remember it’s “2 minus square root 3”—the smallest standard tan value you’ll see. This can help you avoid mistakes and save time in board and JEE exams.
Example Trick: If tan 75° comes up, instantly recall it’s “2 plus square root 3”—just swap the sign!
- tan 15° = 2 − √3
- tan 75° = 2 + √3
- tan 15° × tan 75° = 1 (useful for checking or practicing quick MCQs)
Vedantu’s live sessions include many such shortcuts to help you build speed and accuracy for Olympiad, Boards, and JEE.
Try These Yourself
- What is the value of tan 75°? Can you use symmetry to find it?
- Solve: Find the exact value of tan 105° using subtraction/addition formulas.
- Write tan 15° in decimal up to 4 digits.
- Without calculator, which is larger: tan 15° or tan 30°?
- What is tan 15° + tan 75°?
Frequent Errors and Misunderstandings
- Rushing with the formula and missing the negative/positive sign while rationalizing.
- Forgetting to use tan(A − B) and incorrectly adding values instead of subtracting.
- Mixing up values of tan 15° with tan 30° or tan 45° during MCQs.
Relation to Other Concepts
The idea of value of tan 15 connects closely with topics such as Trigonometric Identities and Trigonometric Ratios of Standard Angles. Mastering this helps with understanding more advanced concepts like compound angles and heights and distances—core areas for board exams and entrance tests.
Classroom Tip
A quick way to remember value of tan 15: Recite “2 minus root 3, tan fifteen comes to me.” Vedantu’s teachers often use such rhymes and visualizations in classes and board revision sessions to boost memory retention.
We explored value of tan 15—from definition, formula, step-by-step derivation, memory hacks, frequent mistakes, and its vital role in trigonometry and competitive math. Continue practicing with Vedantu to become confident in solving all such trigonometric value problems—be ready for both class and exams!
Related Topics:
• Trigonometric Identities (for subtraction/addition formula tricks)
• Sin 30 Degrees (often used with tan(A−B))
• Value of Tan 45 Degrees (quick reference for tan 45 used in derivation)
• Trigonometry Table (for all sin, cos, tan standard values)
• Tan 60 Degrees (comparison/identity practice)
FAQs on Tan 15 Degrees: Exact Value, Formula, and Stepwise Derivation
1. What is the exact value of tan 15 degrees in root or fraction?
The exact value of tan 15° is 2 - √3, which can also be expressed as a fraction: (√3 - 1) / (√3 + 1). This simplifies to the surd form 2 - √3, approximately equal to 0.2679.
2. How to calculate tan 15 using trigonometric identities?
We can calculate tan 15° using the tangent subtraction formula: tan(A - B) = (tan A - tan B) / (1 + tan A tan B). By setting A = 45° and B = 30°, we get:
- tan 45° = 1
- tan 30° = 1/√3
Substituting these values into the formula gives:
tan 15° = (1 - 1/√3) / (1 + 1/√3) = (√3 - 1) / (√3 + 1) = 2 - √3
3. Is tan 15 degrees a rational number?
No, tan 15° is an irrational number because it involves the square root of 3 (√3), which is irrational. Irrational numbers cannot be expressed as a simple fraction of two integers.
4. How do I use tan 15 in trigonometry problems and MCQs?
Knowing the value of tan 15° can simplify trigonometric calculations. In MCQs, you can often use it to directly substitute into equations or to simplify expressions. It's also useful when dealing with angles that can be expressed as sums or differences involving 15°, 30°, 45°, etc. Memorizing the value (2 - √3) is beneficial for quick problem-solving.
5. What is the sum of tan 15 and cot 15?
The sum of tan 15° and cot 15° is 4. This is because tan θ + cot θ = 2/sin(2θ), and when θ = 15°, 2θ = 30°, so sin(30°) = 1/2, resulting in tan 15° + cot 15° = 4.
6. Why is tan 15° useful in geometry and coordinate proofs?
The value of tan 15° appears in various geometric problems and coordinate proofs, particularly when dealing with triangles with angles of 15°, 75°, and 90°. Its exact value aids precise calculation and simplification in geometric derivations.
7. How does knowing tan 15° help with mental calculation tricks?
Knowing the value of tan 15° (2 - √3) allows for quicker calculations when solving trigonometric equations that include 15° or multiples thereof. This is especially valuable in competitive exams where speed and accuracy are essential.
8. Can you derive tan 15 using the sine and cosine addition formulas?
Yes, you can derive tan 15° using the sine and cosine addition formulas. Since tan θ = sin θ / cos θ, calculate sin 15° and cos 15° separately using these formulas and then find their ratio to obtain tan 15°.
9. How does the value of tan 15 compare with tan 75 and tan 150?
tan 75° is the reciprocal of tan 15° (approximately 3.732), while tan 150° is equal to -tan 30° which is -1/√3 (approximately -0.577). The relationship is based on the trigonometric identities and the property of the tangent function.
10. In which standard exams is the value of tan 15° most frequently asked?
The value of tan 15° is frequently tested in various standard exams such as the CBSE board exams (class 10 and 12), competitive entrance exams like JEE Main and Advanced, and other undergraduate mathematics examinations. It's a fundamental trigonometric value and its application is important for problem solving and understanding.
11. What are some common mistakes students make when calculating tan 15°?
Common mistakes include incorrectly applying the tangent subtraction formula, making errors in simplifying surds, and forgetting that tan 15° is an irrational number. Double-checking calculations and using a calculator to verify intermediate results can be helpful.
