Tan 5pi/4 = 1: Exact Value & Unit Circle

What Is The Value Of Tan 5pi/4?

Tan 5pi/4 is exactly $1$. In symbols, $\tan\frac{5\pi}{4} = 1$, and to four decimal places that is $1.0000$.

The angle can be written two ways, and both point to the same place on a circle:

• Radians: $\dfrac{5\pi}{4}$

• Degrees: $\dfrac{5\pi}{4} \times \dfrac{180^\circ}{\pi} = 225^\circ$

So $\tan\frac{5\pi}{4} = \tan 225^\circ = 1$. This is an exact value, not a rounded one. Unlike an angle such as $37^\circ$, the angle $225^\circ$ is built from the friendly $45^\circ$ family, so its tangent comes out to a clean whole number rather than a messy decimal. If you want the underlying ratio itself, the tangent function is defined as $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, and that ratio is what we evaluate below.

How Do You Find Tan 5pi/4?

There is a two-part recipe for any tangent of a special angle: find the reference angle, then fix the sign from the quadrant. Both steps are short.

Step 1: Find the reference angle. The reference angle is the acute angle between the terminal side and the horizontal axis. Since $225^\circ$ sits past $180^\circ$, subtract:

$$225^\circ - 180^\circ = 45^\circ \qquad \Longleftrightarrow \qquad \frac{5\pi}{4} - \pi = \frac{\pi}{4}$$

The reference angle is $\frac{\pi}{4}$ ($45^\circ$), and $\tan\frac{\pi}{4} = 1$. You can confirm that base value at tan pi/4 or, in degree form, at tan 45 degrees.

Step 2: Fix the sign from the quadrant. The angle $225^\circ$ lands in the third quadrant (between $180^\circ$ and $270^\circ$). The ASTC rule, remembered as All Students Take Calculus or the CAST diagram, tells you which ratio is positive in each quadrant:

• Quadrant I: All positive.

• Quadrant II: Sine positive.

• Quadrant III: Tangent positive.

• Quadrant IV: Cosine positive.

The third quadrant is the "T" quadrant, so tangent is positive there. Combine the two steps: the reference value is $1$ and the sign is positive, so

$$\tan\frac{5\pi}{4} = +\tan\frac{\pi}{4} = 1.$$

Where Does 5pi/4 Sit On The Unit Circle?

On the unit circle, an angle is measured anticlockwise from the positive $x$-axis, and the point where the terminal side meets the circle has coordinates $(\cos\theta, \sin\theta)$. Tangent is then the ratio $\dfrac{\sin\theta}{\cos\theta}$, which is the $y$-coordinate divided by the $x$-coordinate.

At $\frac{5\pi}{4}$ the terminal side points down and to the left, exactly halfway through the third quadrant, and it meets the circle at

$$\left(\cos\frac{5\pi}{4},\ \sin\frac{5\pi}{4}\right) = \left(-\frac{\sqrt{2}}{2},\ -\frac{\sqrt{2}}{2}\right) \approx (-0.7071,\ -0.7071).$$

Both coordinates are negative, which is the whole story for the sign. Dividing them:

$$\tan\frac{5\pi}{4} = \frac{\sin\frac{5\pi}{4}}{\cos\frac{5\pi}{4}} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1.$$

The two minus signs cancel. That cancellation is exactly why a third-quadrant angle can have a positive tangent even though both of its coordinates are negative. For the separate coordinate values, see sin 5pi/4 and cos 5pi/4.

How Do You Find Tan 5pi/4 From The Right Triangle?

The unit circle explains the sign, but the number $1$ comes straight from a right triangle, and seeing both keeps tangent from splitting into "the wave thing" and "the triangle thing" in your memory.

Drop the reference triangle for $\frac{5\pi}{4}$ into the third quadrant. Its acute angle is the reference angle $45^\circ$, so it is a 45-45-90 triangle, and in that triangle the two legs are equal in length. Tangent of an acute angle is the opposite side over the adjacent side:

$$\tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1.$$

That gives the size of the answer. The quadrant then supplies the sign, and since tangent is positive in Quadrant III, the triangle value carries through unchanged:

$$\tan\frac{5\pi}{4} = 1.$$

Both roads, the unit circle and the right triangle, land on the same value. A cleaner way to say it: tangent measures a slope, the ratio of vertical change to horizontal change, and when the rise equals the run the slope is $1$.

What Are The Related Tangent Values Around The Unit Circle?

The angle $\frac{5\pi}{4}$ belongs to the $45^\circ$ family, the four angles whose reference angle is $\frac{\pi}{4}$. Their tangents are all $\pm 1$, and the sign just tracks the quadrant.

Table: The $45^\circ$-family angles, their coordinates, and their tangents.

Notice $\tan\frac{5\pi}{4} = \tan\frac{\pi}{4} = 1$. That is the period of tangent at work: tangent repeats every $\pi$ radians ($180^\circ$), so adding $\pi$ to $\frac{\pi}{4}$ leaves the tangent unchanged. Compare the negative cases at tan 3pi/4, and see the next third-quadrant tangent at tan 7pi/6. A full sweep of these values lives in the trigonometric table.

Why Is Tan 5pi/4 Equal To 1?

The value is not a coincidence of the number line. Three facts, stacked, force it.

• The reference angle is $45^\circ$. Halfway through the third quadrant is exactly $45^\circ$ past the horizontal, and the tangent of $45^\circ$ is $1$ because its triangle has equal legs.

• Both coordinates are negative. At $225^\circ$ the point is $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$, so sine and cosine are both negative.

• Negative over negative is positive. Tangent is $\frac{\sin}{\cos}$, and dividing two equal negatives gives $+1$.

Put simply, tangent asks "how steep is the line from the origin to this point?" The line to $225^\circ$ has the same steepness as the line to $45^\circ$, since one is just the other extended straight through the origin. Same line, same slope, so the same tangent.

Who Discovered The Tangent Function?

Tangent did not begin as a ratio in a triangle. It began as a shadow. Ancient astronomers measured the length of the shadow a vertical stick (a gnomon) cast at different sun heights, and that shadow length is exactly the tangent of the sun's angle. The Latin name for the straight shadow, umbra recta, was the tangent's first identity.

Two more figures shaped the ratio we now write as $\tan$:

• Al-Battani (858–929, Mesopotamia) used shadow ratios throughout his astronomy and produced tables that fed directly into later tangent tables.

• Thomas Fincke (1561–1656, Denmark) gave the function its modern name, printing the word "tangent" in his 1583 book Geometria Rotundi.

Where Is Tan 5pi/4 Used In The Real World?

A tangent of $1$ is the mathematics of "as much up as across," so a $45^\circ$ slope (and its mirror at $225^\circ$) shows up wherever a perfectly balanced incline or bearing matters.

• Ramps and roofs: a ramp or roof pitch where the rise equals the run sits at $\tan = 1$; builders call this a "one-in-one" slope, the steepest a comfortable staircase approaches.

• Navigation and radar: a heading pointing equally south and west (into the third quadrant of a compass grid) is a $225^\circ$ bearing, and its tangent sets how the east-west and north-south components relate.

• Computer graphics: rotating a sprite or camera to a $225^\circ$ diagonal uses the sine and cosine at $\frac{5\pi}{4}$, and their ratio, the tangent, fixes the on-screen slope of the movement.

• Physics of projectiles and forces: resolving a force aimed down-and-left into horizontal and vertical parts leans on the same $\frac{5\pi}{4}$ coordinates.

The thread running through all of these is slope. Tangent turns an angle into a rate of change, which is why it travels far beyond the triangle it was born in.

What Are The Most Common Mistakes With Tan 5pi/4?

These three slips account for most wrong answers on this angle, drawn from the recurring sign-and-mode confusion seen across student Q&A threads for "tan(5pi/4)."

Assuming the tangent is negative because both coordinates are negative.

Where it slips in:

A student sees that sine and cosine are both negative in the third quadrant and concludes the tangent must be negative too.

Don't do this:

Do not carry the minus sign into the ratio without dividing. Two negatives do not stay negative when one is divided by the other.

The correct way:

Divide the signs through: $\dfrac{-}{-} = +$. In Quadrant III tangent is positive (the "T" in ASTC), so $\tan\frac{5\pi}{4} = +1$.

Leaving the calculator in the wrong angle MODE.

Where it slips in:

A student types $\tan(5\pi/4)$ with the calculator set to degrees, or types $\tan(225)$ with it set to radians, and reads off a strange decimal.

Don't do this:

Do not trust the display until the MODE matches the angle. $\frac{5\pi}{4}$ is a radian measure; $225$ is a degree measure. They must not be mixed.

The correct way:

Set the calculator to radians for $\tan(5\pi/4)$, or to degrees for $\tan(225^\circ)$. Both then return $1$.

Misreading the reference angle.

Where it slips in:

A student subtracts from the wrong benchmark, using $270^\circ - 225^\circ = 45^\circ$ by luck, or $225^\circ - 90^\circ$, and loses track of which quadrant they are in.

Don't do this:

Do not guess the reference angle. In the third quadrant the reference angle is measured from $180^\circ$, not from $90^\circ$ or $270^\circ$.

The correct way:

For a Quadrant III angle, subtract $180^\circ$ (or $\pi$): $225^\circ - 180^\circ = 45^\circ$, giving reference angle $\frac{\pi}{4}$.

Practice Problems On Tan 5pi/4

Work each one, then check against the answer. A blank unit circle beside you helps.

1. State $\tan\frac{5\pi}{4}$ in exact form and as a decimal to 4 dp.
   (Answer: $1$; $1.0000$.)

2. Convert $\frac{5\pi}{4}$ radians to degrees.
   (Answer: $225^\circ$.)

3. What is the reference angle of $\frac{5\pi}{4}$?
   (Answer: $\frac{\pi}{4}$, or $45^\circ$.)

4. Using $\tan\theta = \frac{\sin\theta}{\cos\theta}$, evaluate the ratio from the coordinates $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$.
   (Answer: $\frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1$.)

5. Is $\tan\frac{5\pi}{4}$ positive or negative, and which ASTC quadrant explains it?
   (Answer: Positive; Quadrant III, the "T" quadrant.)

6. Use the period of tangent to find another angle with the same value, between $0$ and $2\pi$.
   (Answer: $\frac{\pi}{4}$, since tangent repeats every $\pi$.)

Where Should You Go Next After Tan 5pi/4?

You now have a repeatable method: reference angle, then quadrant sign. Three doors carry it further.

1. Unit circle with tangent. See how the tangent value behaves for every angle around the circle, not just this one, and where it shoots to infinity.

2. Tan 3pi/4. The second-quadrant partner of this angle, where the same reference angle produces $-1$ instead of $+1$.

3. Trigonometric ratios in radians. Lock in the radian habit so $\frac{5\pi}{4}$ reads as naturally as $225^\circ$.

If your child is building this fluency, a live Bhanzu trainer teaches special-angle values starting from the "why", the reference angle and the quadrant sign, rather than memorised tables, in the Bhanzu trigonometry program.