What Is The Value Of Cos 225 Degrees?
Cos 225 Degrees is $-\frac{\sqrt{2}}{2}$, an exact value that also appears as $-\frac{1}{\sqrt{2}}$ and rounds to $-0.7071$ (to four decimal places).
$$\cos 225^\circ = -\frac{\sqrt{2}}{2} = -\frac{1}{\sqrt{2}} \approx -0.7071$$
The two surd forms are equal. Rationalising $-\frac{1}{\sqrt{2}}$ by multiplying top and bottom by $\sqrt{2}$ gives $-\frac{\sqrt{2}}{2}$, so both write the same number.
The angle itself has two standard names. In degrees it is $225^\circ$, and in radians it is $\frac{5\pi}{4}$, since $225 \times \frac{\pi}{180} = \frac{5\pi}{4} \approx 3.9270$. If the radian form is new, the page on what is a radian sets it up from scratch.
How Do You Find Cos 225 Degrees?
The quickest route uses two ideas: the reference angle and the sign for the quadrant. Both come from where the angle sits.
Step 1, locate the quadrant. Since $180^\circ < 225^\circ < 270^\circ$, the angle is in the third quadrant.
Step 2, find the reference angle. The reference angle is the gap from the nearest part of the $x$-axis: $225^\circ - 180^\circ = 45^\circ$.
Step 3, fix the sign. Under the CAST (or ASTC) rule, only tangent is positive in the third quadrant, so cosine there is negative.
Step 4, combine. $\cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2}$.
The reference-angle rule works because $225^\circ$ and $45^\circ$ make the same acute angle with the horizontal axis. Their cosine values match in size, and only the sign changes with the quadrant. The base value $\cos 45^\circ = \frac{\sqrt{2}}{2}$ is worked out in full on cos 45 degrees.
How Do You Find Cos 225 Degrees Using The Angle-Sum Formula?
A second method proves the same value from an identity, so the two answers can be checked against each other. Write $225^\circ$ as $180^\circ + 45^\circ$ and apply the cosine angle-sum rule $\cos(A + B) = \cos A \cos B - \sin A \sin B$.
$$\cos 225^\circ = \cos(180^\circ + 45^\circ)$$
$$= \cos 180^\circ \cos 45^\circ - \sin 180^\circ \sin 45^\circ$$
$$= (-1)\left(\frac{\sqrt{2}}{2}\right) - (0)\left(\frac{\sqrt{2}}{2}\right)$$
$$= -\frac{\sqrt{2}}{2}$$
The same value drops out. A related split, $225^\circ = 270^\circ - 45^\circ$, gives the identical answer through the difference formula. The full expansion of these identities lives on cos a minus b formula.
Where Does 225 Degrees Sit On The Unit Circle?
On the unit circle, cosine is the $x$-coordinate of the point where the angle's arm meets the circle. At $225^\circ$ that point is $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$, sitting in the lower-left. The $x$-coordinate reads off the cosine directly.
$$225^\circ = \frac{5\pi}{4} \quad\Rightarrow\quad \left(\cos 225^\circ,\ \sin 225^\circ\right) = \left(-\frac{\sqrt{2}}{2},\ -\frac{\sqrt{2}}{2}\right)$$
Both coordinates are equal and both are negative, which is why $\cos 225^\circ$ and $\sin 225^\circ$ share the value $-\frac{\sqrt{2}}{2}$. You can see the same value from a right triangle: drop a perpendicular from the circle point to the $x$-axis, and the little triangle has two equal legs (the $45^\circ$ reference angle), so the horizontal leg has length $\frac{\sqrt{2}}{2}$, carried into negative territory by the quadrant.
For an interactive version of this picture, the unit circle with tangent page lets you move the angle and watch the coordinates change.
What Are The Related Values Around Cos 225 Degrees?
The angle $225^\circ$ belongs to a family that all share the $45^\circ$ reference angle, one per quadrant. Their cosines are equal in size and differ only in sign, which makes the pattern easy to remember.
Table: The 45° reference-angle family, showing how the sign of cosine changes by quadrant.
Angle | Radians | Cosine | Sine | Tangent |
|---|---|---|---|---|
$45^\circ$ | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2} \approx 0.7071$ | $\frac{\sqrt{2}}{2} \approx 0.7071$ | $1$ |
$135^\circ$ | $\frac{3\pi}{4}$ | $-\frac{\sqrt{2}}{2} \approx -0.7071$ | $\frac{\sqrt{2}}{2} \approx 0.7071$ | $-1$ |
$225^\circ$ | $\frac{5\pi}{4}$ | $-\frac{\sqrt{2}}{2} \approx -0.7071$ | $-\frac{\sqrt{2}}{2} \approx -0.7071$ | $1$ |
$315^\circ$ | $\frac{7\pi}{4}$ | $\frac{\sqrt{2}}{2} \approx 0.7071$ | $-\frac{\sqrt{2}}{2} \approx -0.7071$ | $-1$ |
The quadrant-II member is detailed on cos 135 degrees, and the radian twin of this angle is on cos 5pi 4, with its partner value on sin 5pi 4. The neighbouring axis values, cos 180 degrees and cos 270 degrees, bracket $225^\circ$ on either side. Angles like $45^\circ$ and $225^\circ$ appear in India's NCERT Class 11 trigonometry and in the US Common Core standard HSF.TF.A.3, so the same value is examined under both systems.
Why Is Cos 225 Degrees Negative?
The sign is not a rule to memorise. It follows from where the angle points.
Cosine is a horizontal position. On the unit circle, cosine measures how far right or left the point is. Rightward is positive, leftward is negative.
225° points to the lower-left. Its $x$-coordinate sits on the negative side of the axis, so the cosine has to be negative.
The size comes from the 45° reference angle. That acute angle fixes the magnitude at $\frac{\sqrt{2}}{2}$, and the quadrant only decides the sign.
So $\cos 225^\circ = -\frac{\sqrt{2}}{2}$ is the leftward reach of a southwest direction. The value pairs naturally with sine, since both coordinates of the $225^\circ$ point are equal, a fact drawn out on sin cos tan.
Who Discovered The Trigonometry Behind Cos 225 Degrees?
Nobody set out to compute $\cos 225^\circ$ on its own. The value is one entry in a much older project: building tables that turn angles into numbers. That work spans continents and centuries.
Two other figures shaped the same tables:
Hipparchus of Nicaea (c. 190–120 BCE, Greece) built the earliest known table of chords, the ancestor of the sine table, to predict the positions of stars.
Madhava of Sangamagrama (c. 1340–1425, India) found the infinite series that let mathematicians compute values like $\cos 45^\circ$ to many decimal places, the method a modern calculator still echoes.
Where Is Cos 225 Degrees Used In The Real World?
Diagonal, third-quadrant values like this one appear wherever motion or direction runs down and to the left.
Navigation and bearings: a heading toward the southwest breaks into a westward (negative) and southward part, and cosine gives the horizontal share.
Alternating current: voltages and currents are modelled as cosine waves, and the value at $\frac{5\pi}{4}$ of the cycle is exactly this negative swing below the axis.
Computer graphics and games: rotating a sprite by $225^\circ$ multiplies its position by $\cos 225^\circ$ and $\sin 225^\circ$, placing it in the lower-left of the screen.
Physics of oscillation: a pendulum or spring past its midpoint sits at a negative displacement that a cosine term describes.
One value, read off a circle, turns up in maps, circuits, screens, and swinging weights. That reach is what makes the special angles worth knowing by heart.
What Are The Most Common Mistakes With Cos 225 Degrees?
Three errors account for most wrong answers on this value. Each one is easy to catch once you know where it hides.
Forgetting cosine is negative in the third quadrant.
Where it slips in:
A student finds the reference angle correctly, gets $\cos 45^\circ = \frac{\sqrt{2}}{2}$, then writes that as the final answer.
Don't do this:
Do not report a positive value. In the third quadrant, cosine is below zero.
The correct way:
Apply the CAST rule first. Only tangent is positive in the third quadrant, so $\cos 225^\circ = -\frac{\sqrt{2}}{2}$, not $+\frac{\sqrt{2}}{2}$.
Reading the angle in the wrong calculator mode.
Where it slips in:
A student types 225 while the calculator is set to radians, and reads off a value near $0.9998$ instead of $-0.7071$.
Don't do this:
Do not trust the display before checking the mode indicator.
The correct way:
Set the calculator to degrees for $225^\circ$, or enter $\frac{5\pi}{4}$ when it is in radian mode. Confirm the result is close to $-0.7071$.
Misidentifying the reference angle.
Where it slips in:
A student subtracts from the wrong axis, using $270^\circ - 225^\circ = 45^\circ$ but pairing it with the sign for the fourth quadrant, or uses $225^\circ$ itself as the reference.
The correct way:
Measure the reference angle from the nearest $x$-axis direction. For $225^\circ$ that is $225^\circ - 180^\circ = 45^\circ$, and the point stays firmly in the third quadrant.
Practice Problems On Cos 225 Degrees
Work each one, then check the answer that follows.
Convert $225^\circ$ to radians.
(Answer: $\frac{5\pi}{4}$.)State $\cos 225^\circ$ as an exact surd and as a decimal to four places.
(Answer: $-\frac{\sqrt{2}}{2}$, which is $-0.7071$.)Find $\sec 225^\circ$, the reciprocal of the cosine.
(Answer: $\frac{1}{\cos 225^\circ} = -\sqrt{2} \approx -1.4142$.)Use $\cos 225^\circ$ and $\sin 225^\circ$ to find $\tan 225^\circ$.
(Answer: $\tan 225^\circ = \frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1$.)Evaluate $\cos^2 225^\circ + \sin^2 225^\circ$.
(Answer: $\frac{1}{2} + \frac{1}{2} = 1$, matching the Pythagorean identity.)Which two other angles in $[0^\circ, 360^\circ)$ share the cosine magnitude $\frac{\sqrt{2}}{2}$?
(Answer: $135^\circ$ shares the sign, while $45^\circ$ and $315^\circ$ give the positive value.)
Where Should You Go Next After Cos 225 Degrees?
A single value opens onto the whole special-angle system. A few natural doors lead onward.
Cos 45 degrees. The positive base value that every $45^\circ$-family angle is built from.
Trigonometric ratios of specific angles. The full grid of sine, cosine, and tangent for the angles you meet most.
Unit circle with tangent. Move the angle yourself and watch cosine change sign as you cross into each quadrant.
If your child is building this foundation, a live Bhanzu trainer teaches special angles starting from the picture (the unit circle and the reference triangle) rather than a memorised table, in the Bhanzu trigonometry program.
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