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How to Work Out Tan 45 Without A Calculator

Reviewed by Calculator Editorial Team

Calculating tan(45) without a calculator is surprisingly simple when you understand the underlying geometry. This guide will show you two reliable methods to determine the tangent of 45 degrees using basic principles.

What is tan(45)?

The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For a 45-degree angle, this relationship creates an isosceles right triangle where the two legs are equal in length.

tan(θ) = opposite / adjacent

For θ = 45°, opposite = adjacent = x

Therefore, tan(45°) = x / x = 1

This fundamental property makes tan(45) equal to 1, a value you can derive without any calculation tools.

Method 1: Using Geometry

This method uses the properties of a square to find tan(45°).

  1. Draw a square with each side measuring 1 unit.
  2. Draw a diagonal from one corner to the opposite corner.
  3. The diagonal divides the square into two right-angled triangles.
  4. Each triangle has two sides of 1 unit (the sides of the square) and the diagonal as the hypotenuse.
  5. According to the Pythagorean theorem: diagonal² = 1² + 1² = 2
  6. Therefore, diagonal = √2
  7. In one of the right-angled triangles, the sides are 1, 1, and √2.
  8. The tangent of the 45° angle is opposite/adjacent = 1/1 = 1.

This method relies on the fact that a square's diagonal creates congruent 45-45-90 triangles, where the legs are equal and the tangent ratio simplifies to 1.

Method 2: Using Trigonometry

This method uses the unit circle and trigonometric identities.

  1. Consider the unit circle where the radius is 1.
  2. At 45° from the positive x-axis, the coordinates are (cos(45°), sin(45°)).
  3. Using trigonometric identities: cos(45°) = sin(45°) = √2/2 ≈ 0.7071
  4. The tangent is defined as sin(θ)/cos(θ).
  5. Therefore, tan(45°) = (√2/2)/(√2/2) = 1

This method shows how trigonometric identities can be used to find tan(45°) when you know the sine and cosine values for 45°.

Verification

To ensure our result is correct, let's verify using a different approach:

  1. Consider a right-angled triangle with angles 30°, 60°, and 90°.
  2. The sides are in the ratio 1 : √3 : 2.
  3. For a 45° angle, we can use the complementary angle property: tan(45°) = cot(45°).
  4. In the 30-60-90 triangle, tan(60°) = √3, and tan(30°) = 1/√3.
  5. Since tan(45°) is the geometric mean of tan(30°) and tan(60°), we can calculate it as √(tan(30°) × tan(60°)) = √((1/√3) × √3) = √1 = 1.

This verification confirms that tan(45°) is indeed equal to 1.

FAQ

Why is tan(45) equal to 1?

tan(45°) equals 1 because in a 45-45-90 triangle, the opposite and adjacent sides are equal in length, making the ratio 1/1 = 1.

Can I use this method for other angles?

Yes, similar geometric methods can be used for other standard angles like 30° and 60°, but tan(45°) is particularly straightforward due to the equal sides in the isosceles right triangle.

Is tan(45) always exactly 1?

Yes, tan(45°) is exactly 1 in all cases because the geometric properties of a 45-45-90 triangle are consistent and well-defined.

What's the difference between tan and cot?

tan(θ) is opposite/adjacent, while cot(θ) is adjacent/opposite. For θ = 45°, tan(45°) = cot(45°) = 1 because the opposite and adjacent sides are equal.