Tan 210 Without Calculator
'/%3E%3Ccircle cx='48' cy='39' r='19' fill='%23f2c9a5'/%3E%3Cpath d='M27 35c3-16 39-17 42 2-8-8-27-9-42-2z' fill='%23374151'/%3E%3Cpath d='M21 86c5-24 49-28 56 0z' fill='%232563eb'/%3E%3Ccircle cx='41' cy='41' r='2' fill='%23111827'/%3E%3Ccircle cx='55' cy='41' r='2' fill='%23111827'/%3E%3Cpath d='M42 52c4 3 9 3 13 0' stroke='%2392400e' stroke-width='3' fill='none' stroke-linecap='round'/%3E%3C/svg%3E)
Reviewed by Calculator Editorial Team
Calculating tan 210° without a calculator requires understanding trigonometric identities and reference angles. This guide explains the method, provides a step-by-step calculation, and includes a practical example.
How to calculate tan 210° without a calculator
The tangent of 210° can be found using trigonometric identities and reference angles. Here's how to do it:
- Recognize that 210° is in the third quadrant where tangent is positive.
- Find the reference angle by subtracting 180° from 210°: 210° - 180° = 30°.
- Use the tangent of the reference angle (tan 30°).
- Apply the appropriate sign based on the quadrant.
Remember that in the third quadrant, both sine and cosine are negative, but their ratio (tangent) is positive.
The formula for tan 210°
The tangent of an angle in the third quadrant can be calculated using the reference angle:
tan(θ) = tan(θ - 180°)
For θ = 210°:
tan(210°) = tan(30°)
We know that tan(30°) = √3/3 ≈ 0.577.
Step-by-step calculation
- Identify the quadrant: 210° is in the third quadrant (180°-270°).
- Calculate the reference angle: 210° - 180° = 30°.
- Recall that tan(30°) = √3/3 ≈ 0.577.
- Since tangent is positive in the third quadrant, tan(210°) = tan(30°).
The exact value of tan(210°) is √3/3, and the approximate value is 0.577.
Worked example
Let's calculate tan(210°) using the method above:
- 210° is in the third quadrant.
- Reference angle = 210° - 180° = 30°.
- tan(30°) = √3/3 ≈ 0.577.
- Therefore, tan(210°) = √3/3 ≈ 0.577.
The exact value is √3/3, and the decimal approximation is 0.577.
FAQ
- Why is tan(210°) positive?
- Because 210° is in the third quadrant where both sine and cosine are negative, but their ratio (tangent) is positive.
- What is the reference angle for 210°?
- The reference angle is 30° (210° - 180°).
- Can I use this method for other angles?
- Yes, this method works for any angle in the third quadrant by finding its reference angle.
- What is the exact value of tan(210°)?
- The exact value is √3/3.
- How accurate is the decimal approximation?
- The decimal approximation 0.577 is accurate to three decimal places.