How to Find Tan of 75 Without Calculator

Calculating the tangent of 75 degrees without a calculator requires understanding trigonometric identities and applying them systematically. This guide explains the process step-by-step, including the use of reference angles and the tangent addition formula.

Understanding Tangent

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. In the unit circle, tan(θ) represents the y-coordinate divided by the x-coordinate of the corresponding point.

For angles outside the standard 30°, 45°, 60°, and 90° values, we can use trigonometric identities to find their tangent values. The tangent of 75 degrees is particularly useful in various geometric and engineering applications.

Calculating Tan(75°)

To find tan(75°), we can use the tangent addition formula:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

We know that 75° can be expressed as 45° + 30°. Therefore, we can calculate tan(75°) using the known values of tan(45°) and tan(30°).

tan(45°) = 1

tan(30°) = √3/3 ≈ 0.577

Step-by-Step Method

Identify that 75° = 45° + 30°.
Apply the tangent addition formula:

tan(75°) = tan(45° + 30°) = (tan 45° + tan 30°) / (1 - tan 45° tan 30°)
Substitute the known values:

tan(75°) = (1 + √3/3) / (1 - 1 * √3/3)
Simplify the expression:

tan(75°) = (3 + √3) / (3 - √3)
Rationalize the denominator by multiplying numerator and denominator by the conjugate of the denominator:

tan(75°) = [(3 + √3)(3 + √3)] / [(3 - √3)(3 + √3)]
Calculate the numerator and denominator:

Numerator: (3 + √3)² = 9 + 6√3 + 3 = 12 + 6√3

Denominator: 9 - (√3)² = 9 - 3 = 6
Divide the numerator by the denominator:

tan(75°) = (12 + 6√3) / 6 = 2 + √3

The exact value of tan(75°) is 2 + √3, which is approximately 3.732.

Verification

To ensure the accuracy of our calculation, we can verify the result using the sine and cosine values of 75°.

We know that:

tan(θ) = sin(θ) / cos(θ)

Using the sine and cosine addition formulas:

sin(75°) = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°

cos(75°) = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30°

Substituting the known values:

sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4) = (√6 + √2)/4

cos(75°) = (√2/2)(√3/2) - (√2/2)(1/2) = (√6/4) - (√2/4) = (√6 - √2)/4

Now, calculate tan(75°):

tan(75°) = sin(75°) / cos(75°) = [(√6 + √2)/4] / [(√6 - √2)/4] = (√6 + √2) / (√6 - √2)

Rationalizing the denominator:

tan(75°) = [(√6 + √2)(√6 + √2)] / [(√6 - √2)(√6 + √2)] = (6 + 2√12 + 2) / (6 - 2) = (8 + 4√3) / 4 = 2 + √3

This confirms our earlier result of tan(75°) = 2 + √3.

Practical Applications

Knowing how to find tan(75°) without a calculator is valuable in various fields:

Engineering: Calculating angles in structural designs and mechanical systems.
Architecture: Determining roof slopes and other geometric measurements.
Physics: Solving problems involving waves, oscillations, and other periodic phenomena.
Mathematics: Exploring trigonometric identities and their applications.

Understanding the exact value of tan(75°) allows for precise calculations in these domains.

FAQ

What is the exact value of tan(75°)?

The exact value of tan(75°) is 2 + √3.

How can I remember the exact value of tan(75°)?

You can derive it using the tangent addition formula with tan(45°) and tan(30°).

Is tan(75°) the same as tan(105°)?

No, tan(105°) is negative because 105° is in the second quadrant where tangent is negative.

Can I use a calculator to verify the exact value?

Yes, a calculator can confirm that tan(75°) ≈ 3.732, which matches the exact value of 2 + √3.

Where is tan(75°) used in real life?

It's used in engineering, architecture, and physics for calculating angles and solving geometric problems.