Tan 55 Degrees — Value of tan(55°) and How to Find It

Tan 55 Degrees — Value of tan(55°) and How to Find It

TL;DR

The value of tan 55 degrees is approximately 1.4281 — it is not a special-angle exact value, so there is no clean surd for it. This article shows how to find tan⁡55° honestly (calculator, the cofunction cot⁡35°, and table interpolation), gives the radian form, and explains why the value is greater than 1.

The value of tan 55 degrees is approximately 1.4281 (1.42814801 to eight places). Unlike tan⁡45° or tan⁡60°, the angle 55° is not a special angle, so tan⁡55° has no simple exact surd — it is read from a calculator, a trig table, or rewritten as the cofunction cot⁡35°.

Quick Answer:

Result: tan⁡55°≈1.4281 In radians: tan⁡(11π/36)=tan(0.95993)≈1.4281 Notation: decimal approximation — 1.42814801 (8 dp) Method shown: calculator (degree mode), the cofunction identity tan⁡55°=cot⁡35°, and table interpolation Exact form: none simple — 55° is not a special angle, so no clean radical exists

What Does Tan 55 Degrees Mean?

Tangent of an angle is the ratio of sine to cosine: tan⁡θ=sin⁡θ/cos⁡θ. On the unit circle, that is the y-coordinate divided by the x-coordinate of the point at angle θ.

A quadrant is one of the four regions the axes divide the plane into, numbered anticlockwise from the top right; 55° lands in Quadrant I, where sine and cosine are both positive, so tangent is positive too.

Because 55° is past 45° (where sin⁡=cos), the numerator now exceeds the denominator, so tan⁡55°>1. That ratio works out to about 1.4281.

How Do You Find the Value of Tan 55 Degrees?

Because 55° is not a special angle, there is no surd to simplify to. So how do you find tan 55 degrees without a calculator? You rewrite it as a cofunction or interpolate from a table — here are the three honest routes.

Method 1: Calculator (set to degree mode)

Type tan⁡(55) with the calculator in DEG mode.

tan⁡55°=1.42814801…≈1.4281

In radian mode the same keystrokes give tan⁡(55 rad)≈−6.40 — a completely different number, so the mode matters.

Method 2: Cofunction identity

Tangent and cotangent are cofunctions: tan⁡θ=cot⁡(90°−θ).

tan⁡55°=cot⁡(90°−55°)=cot⁡35°= \frac{1}{tan⁡35°}.

Since tan⁡35°≈0.7002, this gives \frac{1}{0.7002}≈1.4281 — the same value, confirmed a second way.

Method 3: Table interpolation

If a trig table lists tan⁡54°=1.3764 and tan⁡56°=1.4826, estimate tan⁡55° by linear interpolation:

tan⁡55°≈1.3764+\frac{55−54}{56−54}(1.4826−1.3764)=1.3764+0.5(0.1062)=1.4295.

That lands within 0.001 of the true 1.4281. Interpolation carries a slightly larger error for tangent than for sine.

What is tan 55 degrees in radians?

The angle converts to 11π/36≈0.9599, but the value of the tangent is the same number, ≈1.4281. Converting the angle does not change the tangent.

Examples Using Tan 55 Degrees

Example 1

State tan⁡55° to four decimal places.

From a calculator in degree mode, tan⁡55°=1.4281.

Example 2 (wrong path first)

Find tan⁡55° from sin⁡55° and cos⁡55°.

Wrong attempt. A student writes tan⁡55°=sin⁡55°×cos⁡55°=0.8192×0.5736=0.4698.

Why it breaks. Tangent is sine divided by cosine, not multiplied: tan⁡θ=\frac{sin⁡θ}{cos⁡θ}. Multiplying gives a number below 1, which can't be right for an angle past 45°.

Correct. tan⁡55°=\frac{sin⁡55°}{cos⁡55°}=\frac{0.8192}{0.5736}=1.4281.

Example 3

A road climbs at 55° to the horizontal. How many metres does it rise over a 20 m horizontal run?

Rise =20×tan⁡55°=20×1.4281=28.56 m.

Example 4

Compare tan⁡55° with tan⁡45°.

tan⁡45°=1; tan⁡55°=1.4281. The extra 10° raises the value by 0.43 — far more than the same 10° would change a sine.

Example 5

Verify tan⁡55°=cot⁡35° on a calculator.

tan⁡55°=1.42815 and cot⁡35°=\frac{1}{tan⁡35°}=1.42815 — identical, confirming the cofunction identity.

Tan 55 Degrees — Tripping Points to Avoid

Most errors on a non-special tangent come from a few repeatable habits.

Mistake 1: Multiplying sine and cosine instead of dividing

Where it slips in: building tangent from sin⁡θ and cos⁡θ.

Don't do this: writing tan⁡55°=sin⁡55°×cos⁡55°.

The correct way: tangent is the quotient \frac{sin⁡θ}{cos⁡θ}. The habit that fixes this is to read "tangent" as "sine over cosine" before writing anything.

Mistake 2: Using the wrong cofunction

Where it slips in: rewriting tan⁡55° as a complementary angle.

Don't do this: writing tan⁡55°=tan⁡35°.

The correct way: the complement of tangent is cotangent — tan⁡55°=cot⁡35°, which equals \frac{1}{tan⁡35°}, not tan⁡35° itself.

Mistake 3: Forgetting the calculator's angle mode

Where it slips in: the calculator was left in radian mode.

Don't do this: reading tan⁡(55)=−6.40 and reporting it as tan⁡55°.

The correct way: check DEG mode for tan⁡55°; −6.40 is tan⁡(55 radians).

Key Takeaways