What Is The Value Of Sin 225 Degrees?
Sin 225 Degrees is $-\frac{\sqrt{2}}{2}$, which equals $-\frac{1}{\sqrt{2}}$ and rounds to $-0.7071$ (to four decimal places). The value is negative because $225^\circ$ lands in the third quadrant, where the sine of an angle is below zero.
The angle can be written two ways, and both give the same result:
In degrees: $\sin 225^\circ = -\dfrac{\sqrt{2}}{2}$.
In radians: $225^\circ = \dfrac{5\pi}{4}$, so $\sin\dfrac{5\pi}{4} = -\dfrac{\sqrt{2}}{2}$.
To move between the two forms, multiply degrees by $\frac{\pi}{180}$: $225 \times \frac{\pi}{180} = \frac{5\pi}{4}$. A fuller walkthrough lives at converting degrees to radians and what is a radian.
How Do You Find Sin 225 Degrees?
Finding the value takes two decisions: which quadrant the angle sits in (that fixes the sign), and what the reference angle is (that fixes the size). Both come from the same picture.
Step 1: Place the angle and read its sign. An angle of $225^\circ$ is past $180^\circ$ but short of $270^\circ$, so its terminal side falls in the third quadrant. The memory aid ASTC (All, Sine, Tangent, Cosine) marks which ratio is positive in each quadrant. Quadrant III belongs to Tangent, so sine there is negative.
Step 2: Find the reference angle. The reference angle is the acute angle between the terminal side and the horizontal axis. For an angle in Quadrant III, subtract $180^\circ$:
$$225^\circ - 180^\circ = 45^\circ$$
Step 3: Attach the sign to the reference value. The reference angle is $45^\circ$, and $\sin 45^\circ = \frac{\sqrt{2}}{2}$. Because sine is negative in Quadrant III:
$$\sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2} \approx -0.7071$$
That is the whole method for any special angle: quadrant sets the sign, reference angle sets the number. You can check the reference value on the table of specific-angle ratios or the full trigonometric table.
Where Does 225° Sit On The Unit Circle?
On the unit circle (radius $1$, centred at the origin), the sine of an angle is the $y$-coordinate of the point where the angle's terminal side meets the circle. The angle $\frac{5\pi}{4}$ points into the lower-left quadrant, so both coordinates are negative.
The point for $225^\circ$ is:
$$\left(-\frac{\sqrt{2}}{2},; -\frac{\sqrt{2}}{2}\right)$$
The first coordinate is $\cos 225^\circ$ and the second is $\sin 225^\circ$. Reading the height of that point straight off the circle gives $\sin 225^\circ = -\frac{\sqrt{2}}{2}$, matching the reference-angle method.
For a version of this picture that also traces the tangent line, see the unit circle with tangent.
How Do You Read Sin 225 Degrees From A Right Triangle?
The unit-circle point also builds a small right triangle, which ties the value back to the ratio definition of sine. Drop the vertical dashed line from the $225^\circ$ point to the $x$-axis. That line, the piece of the $x$-axis, and the radius form a right triangle whose acute angle at the origin is the $45^\circ$ reference angle.
In that triangle the hypotenuse is the radius, length $1$. The side opposite the $45^\circ$ angle is the vertical drop, whose length is $\frac{\sqrt{2}}{2}$. The ratio definition gives:
$$\sin(\text{reference}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{2}/2}{1} = \frac{\sqrt{2}}{2}$$
The triangle only measures lengths, which are always positive, so it delivers the size $\frac{\sqrt{2}}{2}$. The quadrant then supplies the direction: the point sits below the axis, so the actual coordinate, and therefore $\sin 225^\circ$, is $-\frac{\sqrt{2}}{2}$. The triangle gives the magnitude, the circle gives the sign. For the ratio definitions themselves, see sin cos tan.
Can You Derive Sin 225 Degrees With An Angle-Sum Identity?
Yes. Writing $225^\circ$ as $180^\circ + 45^\circ$ lets the angle-sum identity produce the exact value without a diagram. The identity is $\sin(A + B) = \sin A \cos B + \cos A \sin B$.
$$\sin 225^\circ = \sin(180^\circ + 45^\circ)$$
$$= \sin 180^\circ \cos 45^\circ + \cos 180^\circ \sin 45^\circ$$
$$= (0)\left(\frac{\sqrt{2}}{2}\right) + (-1)\left(\frac{\sqrt{2}}{2}\right)$$
$$= -\frac{\sqrt{2}}{2}$$
The same answer appears from a second split, $225^\circ = 270^\circ - 45^\circ$, using $\sin(A - B) = \sin A \cos B - \cos A \sin B$, since $\sin 270^\circ = -1$ and $\cos 270^\circ = 0$. Both routes confirm $\sin 225^\circ = -\frac{\sqrt{2}}{2}$.
What Are The Sine, Cosine, And Tangent Of 225°?
At $225^\circ$ both coordinates on the unit circle are negative and equal, so sine and cosine match, while their ratio makes tangent positive. This is the pattern for every angle whose reference angle is $45^\circ$.
Table: The $45^\circ$ family across all four quadrants, in degrees and radians.
Angle | Radians | Sine | Cosine | Tangent |
|---|---|---|---|---|
$45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ |
$135^\circ$ | $\frac{3\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-1$ |
$225^\circ$ | $\frac{5\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $1$ |
$315^\circ$ | $\frac{7\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $-1$ |
Reading the $225^\circ$ row: $\sin 225^\circ = -\frac{\sqrt{2}}{2}$, $\cos 225^\circ = -\frac{\sqrt{2}}{2}$, and $\tan 225^\circ = 1$. The tangent is positive because a negative divided by a negative is positive, which is exactly why Quadrant III is the "Tangent" quadrant in ASTC. The radian value $\frac{5\pi}{4}$ has its own reference pages at sin 5π/4 and cos 5π/4, and the base angle at sin 45 degrees.
Why Is Sin 225 Degrees Negative?
The sign is not a rule to memorise, it is a direction on the circle. Sine measures height above or below the horizontal axis, so its sign simply reports whether the point is up or down.
The point is below the axis. At $225^\circ$ the terminal side runs into the lower-left quadrant, so the $y$-coordinate is below zero. A below-zero height is a negative sine.
Quadrant III sends both coordinates down and left. Angles between $180^\circ$ and $270^\circ$ have negative $x$ and negative $y$, so sine and cosine are both negative there.
The size still comes from $45^\circ$. The reference angle fixes how far the point is from the axis, $\frac{\sqrt{2}}{2}$, and the quadrant only decides the sign in front of it.
Sine repeats every $360^\circ$ (or $2\pi$), so $\sin 225^\circ$, $\sin 585^\circ$, and $\sin(-135^\circ)$ all equal $-\frac{\sqrt{2}}{2}$. The circle is the reason the same value keeps returning.
Who Discovered The Sine Values Behind Sin 225 Degrees?
The number $-\frac{\sqrt{2}}{2}$ did not arrive with a sine key on a calculator. People built tables of these values by hand, one angle at a time, across more than a thousand years and three continents.
Two later mathematicians reshaped that work into the sine we use:
Aryabhata (476–550 CE, India) tabulated the half-chord, which he called jya, in his Aryabhatiya around 500 CE. The word travelled through Arabic and Latin and became "sine."
Al-Battani (858–929 CE, Mesopotamia) refined the sine tables and used them for sharp astronomical predictions, carrying the idea from India into the wider mathematical world.
Where Is Sin 225 Degrees Used In The Real World?
A negative sine is not an abstraction. Anywhere something swings, oscillates, or rotates past a centre line, angles like $225^\circ$ describe the part of the cycle that dips below zero.
Alternating current: household mains voltage follows a sine wave, and at a phase of $225^\circ$ (or $\frac{5\pi}{4}$) the voltage is negative, meaning the current is flowing the other way through the circuit.
Sound and waves: a pure tone or a water wave is a sine curve, and $225^\circ$ marks a point in the trough, below the resting level.
Pendulums and springs: the displacement of a swinging or bouncing object is modelled as a sine of the phase angle, so $225^\circ$ is a moment when the object sits on the far, below-centre side of its motion.
Computer graphics and games: rotating a point around a circle uses sine and cosine of the angle, and negative values place the point in the lower-left of the screen.
Navigation and engineering: bearings, forces, and rotations resolved into components use the sine of the direction angle, sign included, to point a quantity the correct way.
The same value that looks like plain homework is what tells a circuit which way the current runs. Mathematics quietly runs the machines around the reader.
What Are The Most Common Mistakes With Sin 225 Degrees?
These four errors account for most lost marks on Quadrant III angles, and each appears in the worked solutions and quadrant notes on the top-ranking pages for this angle.
Giving the answer a positive sign.
Where it slips in:
A student finds the reference value $\sin 45^\circ = \frac{\sqrt{2}}{2}$ and writes it as the final answer, forgetting to check the quadrant.
Don't do this:
Do not report $\sin 225^\circ = \frac{\sqrt{2}}{2}$. The reference value is only the size, never the whole answer.
The correct way:
Check the quadrant before writing the sign. Angle $225^\circ$ is in Quadrant III, where sine is negative (ASTC), so $\sin 225^\circ = -\frac{\sqrt{2}}{2}$.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types sin(225) with the calculator set to radians and reads off roughly $-0.6947$, then trusts it.
Don't do this:
Do not evaluate a degree angle in radian mode. In radian mode, $225$ is treated as $225$ radians, a completely different angle.
The correct way:
Set the calculator to degree mode for $\sin 225^\circ$, or convert first and enter $\sin\frac{5\pi}{4}$ in radian mode. Both give $-0.7071$.
Taking the wrong reference angle.
Where it slips in:
A student uses $225^\circ$ itself as the reference angle, or subtracts from $90^\circ$ instead of $180^\circ$.
Don't do this:
Do not treat the full angle as the reference angle. The reference angle is always the acute gap to the horizontal axis.
The correct way:
In Quadrant III, subtract $180^\circ$: $225^\circ - 180^\circ = 45^\circ$. That $45^\circ$ is the reference angle whose sine you then sign.
Confusing sine with its related values.
Where it slips in:
A student mixes up $\sin 225^\circ$ with $\cos 225^\circ$ or with the sine of a different Quadrant III angle.
Don't do this:
Do not assume every Quadrant III value is the same. At $225^\circ$ sine and cosine happen to match, but tangent does not, and other angles differ.
The correct way:
Read the coordinate you actually need. Sine is the $y$-coordinate, so $\sin 225^\circ = -\frac{\sqrt{2}}{2}$, while $\tan 225^\circ = 1$. The cofunction and reciprocal identities keep these relationships straight.
Practice Problems On Sin 225 Degrees
Work each one, then check against the answer. Keep exact surd forms where you can, and round decimals to four places.
Convert $225^\circ$ to radians.
(Answer: $225 \times \frac{\pi}{180} = \frac{5\pi}{4}$.)State the reference angle for $225^\circ$ and its quadrant.
(Answer: reference angle $45^\circ$, Quadrant III.)Find $\sin 225^\circ$ as an exact value and a decimal.
(Answer: $-\frac{\sqrt{2}}{2} \approx -0.7071$.)Find $\cos 225^\circ$ and $\tan 225^\circ$.
(Answer: $\cos 225^\circ = -\frac{\sqrt{2}}{2}$, $\tan 225^\circ = 1$.)Use the angle-sum identity on $180^\circ + 45^\circ$ to derive $\sin 225^\circ$.
(Answer: $\sin 180^\circ\cos 45^\circ + \cos 180^\circ\sin 45^\circ = 0 - \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2}$.)Evaluate $\sin 225^\circ + \sin 45^\circ$.
(Answer: $-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0$.)
Where Should You Go Next After Sin 225 Degrees?
Sin 225 Degrees is one entry in the special-angle system, and several natural doors open from here.
Trigonometric ratios of specific angles. The full set of exact values for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$ and their partners around the circle.
The unit circle with tangent. See where every angle's sine, cosine, and tangent come from on one diagram.
Trigonometric ratios in radians. Read the same values with the angle measured as $\frac{5\pi}{4}$ rather than $225^\circ$.
If your child is building these foundations, a live Bhanzu trainer teaches special angles starting from the unit circle in the Bhanzu trigonometry program.
Was this article helpful?
Your feedback helps us write better content