What Is The Value Of Cos 315 Degrees?
The value of Cos 315 Degrees is $\frac{\sqrt{2}}{2}$, an exact surd that also equals $\frac{1}{\sqrt{2}}$ and rounds to $0.7071$ (to four decimal places). Written with the angle in both common units:
$$\cos 315^\circ = \cos\frac{7\pi}{4} = \frac{\sqrt{2}}{2} \approx 0.7071$$
The degree form $315^\circ$ and the radian form $\frac{7\pi}{4}$ describe the exact same rotation. To convert, multiply degrees by $\frac{\pi}{180}$:
$$315^\circ \times \frac{\pi}{180} = \frac{315\pi}{180} = \frac{7\pi}{4}$$
The value is positive, and that single fact is where most errors begin. The sine and tangent of $315^\circ$ are both negative, but the cosine is not. The next sections show why.
How Do You Find The Exact Value Of Cos 315 Degrees?
There are two clean routes to the exact value. Both start by placing the angle and both land on the same surd.
The first uses the reference angle, the acute angle between the terminal arm and the horizontal axis. For $315^\circ$, that gap is $360^\circ - 315^\circ = 45^\circ$.
Method 1: Reference angle.
$$\cos 315^\circ = \cos(360^\circ - 45^\circ)$$
$$\cos(360^\circ - 45^\circ) = \cos 45^\circ$$
$$\cos 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071$$
The identity $\cos(360^\circ - \theta) = \cos\theta$ holds because a cosine reads the horizontal position, and rotating a full turn back by $45^\circ$ lands on the same horizontal distance as $45^\circ$ itself.
Method 2: Angle sum.
Split $315^\circ$ as $270^\circ + 45^\circ$ and apply the cosine addition formula:
$$\cos 315^\circ = \cos(270^\circ + 45^\circ)$$
$$= \cos 270^\circ \cos 45^\circ - \sin 270^\circ \sin 45^\circ$$
$$= (0)\left(\tfrac{\sqrt{2}}{2}\right) - (-1)\left(\tfrac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}$$
Both methods agree. Because the reference angle is $45^\circ$, the value ties straight back to a single square: a right triangle with two equal legs, where the cosine of the base angle is one leg over the hypotenuse, $\frac{1}{\sqrt{2}}$. For the base case, see cos 45 degrees.
Where Does 315 Degrees Sit On The Unit Circle?
On the unit circle, a circle of radius $1$ centred at the origin, an angle is measured anticlockwise from the positive x-axis, and the point where the terminal arm meets the circle has coordinates $(\cos\theta, \sin\theta)$. The cosine is the x-coordinate.
At $315^\circ$, the arm has swept three-quarters of the way round and then $45^\circ$ more, stopping in the fourth quadrant, down and to the right. The point it lands on is:
$$\left(\cos 315^\circ,\ \sin 315^\circ\right) = \left(\frac{\sqrt{2}}{2},\ -\frac{\sqrt{2}}{2}\right) \approx (0.7071,\ -0.7071)$$
The x-coordinate is positive (the point is to the right of centre), so $\cos 315^\circ$ is positive. The y-coordinate is negative (below centre), so $\sin 315^\circ$ is negative. This is the double anchor: the value read off the triangle in the previous section is the very same value read off the unit circle here.
Why Is Cos 315 Degrees Positive?
The sign of any trig ratio depends only on the quadrant its angle lands in. A common memory aid, ASTC (also called CAST), records which ratios are positive where:
Quadrant I ($0^\circ$ to $90^\circ$): All ratios positive (the A in ASTC).
Quadrant II ($90^\circ$ to $180^\circ$): Sine positive only (S).
Quadrant III ($180^\circ$ to $270^\circ$): Tangent positive only (T).
Quadrant IV ($270^\circ$ to $360^\circ$): Cosine positive only (C).
At $315^\circ$ the angle is in Quadrant IV, so cosine is the ratio that stays positive while sine and tangent turn negative. That matches the unit circle exactly: a fourth-quadrant point sits to the right of the origin (positive x, giving a positive cosine) and below it (negative y, giving a negative sine).
So the reference angle $45^\circ$ decides the size of the value, $\frac{\sqrt{2}}{2}$, and the quadrant decides the sign, positive. Size from the reference angle, sign from the quadrant. Keep those two decisions separate and the fourth-quadrant sign trap stops being a trap.
What Are The Related Values Around Cos 315 Degrees?
Every angle whose reference angle is $45^\circ$ shares the same magnitude $\frac{\sqrt{2}}{2}$, and only the signs change by quadrant. Seeing the family together makes the pattern obvious.
Table: The 45° reference-angle family across the four quadrants.
Angle | Radians | Cosine | Sine | Tangent |
|---|---|---|---|---|
$\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ | |
$\frac{3\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $-1$ | |
225° | $\frac{5\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $1$ |
315° | $\frac{7\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-1$ |
The six trig ratios of $315^\circ$ itself follow directly, since each is built from the sine and cosine.
Table: All six trigonometric ratios at 315 degrees.
Function | Exact value | Decimal (4 dp) |
|---|---|---|
$\cos 315^\circ$ | $\frac{\sqrt{2}}{2}$ | $0.7071$ |
$\sin 315^\circ$ | $-\frac{\sqrt{2}}{2}$ | $-0.7071$ |
$\tan 315^\circ$ | $-1$ | $-1.0000$ |
$\cot 315^\circ$ | $-1$ | $-1.0000$ |
$\sec 315^\circ$ | $\sqrt{2}$ | $1.4142$ |
$\csc 315^\circ$ | $-\sqrt{2}$ | $-1.4142$ |
For the full grid of standard angles, see the trigonometric table, and for the radian versions, trigonometric ratios in radians.
Who Discovered The Cosine Function?
Nobody woke up one morning and invented cosine. It grew out of centuries of astronomers measuring chords of circles, long before anyone drew a right triangle and named a ratio.
Two more figures carried the idea forward:
Aryabhata (476 – 550 CE, India) tabulated the half-chord, which he called jya. Through Arabic and Latin translation, jya became our word "sine," and cosine followed as the sine of the complementary angle.
Madhava of Sangamagrama (c. 1340 – c. 1425 CE, India) found that sine and cosine could each be written as an infinite sum of terms with alternating signs and rising powers of the angle, roughly two centuries before similar series appeared in Europe. That discovery is how calculators compute a value like $\cos 315^\circ$ to many decimal places today.
Where Is Cos 315 Degrees Used In The Real World?
A single trig value rarely appears alone in an application, but fourth-quadrant cosines like this one show up wherever something rotates or oscillates.
Alternating current: the voltage in a mains circuit rises and falls as a cosine wave, and engineers read its value at specific phase angles (including angles past $270^\circ$) to time equipment correctly.
Ziplines, ramps, and roofs: the horizontal reach of a sloped cable or beam is its length times the cosine of its angle, so a downward slant in the fourth-quadrant sense uses exactly this kind of value.
GPS and navigation: positions on a sphere are resolved into horizontal and vertical components using sine and cosine, and bearings sweep through all four quadrants.
Computer graphics: rotating a sprite or a 3D model by an angle multiplies its coordinates by sines and cosines, and a rotation of $315^\circ$ is a common step in animation loops.
Sound and signals: any repeating waveform is a sum of sine and cosine pieces, and sampling one at a given phase pulls out a value just like $\cos 315^\circ$.
One number, $\frac{\sqrt{2}}{2}$, quietly helps run power grids, playgrounds, satellites, and screens. The same math reaches across fields that look nothing alike.
What Are The Most Common Mistakes With Cos 315 Degrees?
These three errors account for most of the marks lost on fourth-quadrant cosine questions. Each one is easy to avoid once you have seen it.
Making the answer negative.
Where it slips in:
A student knows $\sin 315^\circ$ and $\tan 315^\circ$ are negative and assumes the cosine must be negative too, writing $-\frac{\sqrt{2}}{2}$.
Don't do this:
Do not copy the sign of one ratio onto another. In Quadrant IV the three ratios do not share a sign.
The correct way:
Check the quadrant first. At $315^\circ$ the angle is in Quadrant IV, where cosine alone is positive, so $\cos 315^\circ = +\frac{\sqrt{2}}{2}$ while sine and tangent are negative.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types $\cos(315)$ expecting degrees, but the calculator is set to radians and returns roughly $-0.0874$, a value that looks nothing like the truth.
Don't do this:
Do not trust a number before checking the angle mode. $315$ radians is a completely different rotation from $315$ degrees.
The correct way:
Set the mode to degrees for $\cos 315^\circ$, or convert first and compute $\cos\frac{7\pi}{4}$ in radian mode. Either gives $0.7071$.
Misreading the reference angle.
Where it slips in:
A student subtracts from the wrong benchmark, taking the reference angle as $315^\circ - 180^\circ = 135^\circ$, or simply using $315^\circ$ itself.
Don't do this:
Do not measure the reference angle from the wrong axis. In Quadrant IV it is measured back from $360^\circ$, not from $180^\circ$.
The correct way:
Use $360^\circ - 315^\circ = 45^\circ$. The reference angle for any fourth-quadrant angle is its distance below a full turn.
Practice Problems On Cos 315 Degrees
Work each one, then check against the answer.
Write $\cos 315^\circ$ in exact form and as a decimal to four places.
(Answer: $\frac{\sqrt{2}}{2} \approx 0.7071$.)State the reference angle and quadrant of $315^\circ$.
(Answer: reference angle $45^\circ$, Quadrant IV.)Convert $315^\circ$ to radians.
(Answer: $\frac{7\pi}{4}$.)Find $\sin 315^\circ$ and explain how its sign differs from $\cos 315^\circ$.
(Answer: $\sin 315^\circ = -\frac{\sqrt{2}}{2}$; in Quadrant IV sine is negative while cosine is positive.)Evaluate $\sec 315^\circ$.
(Answer: $\sec 315^\circ = \frac{1}{\cos 315^\circ} = \sqrt{2} \approx 1.4142$.)Using the angle sum $\cos(270^\circ + 45^\circ)$, verify the value of $\cos 315^\circ$.
(Answer: $\cos 270^\circ\cos 45^\circ - \sin 270^\circ\sin 45^\circ = 0 - (-1)\tfrac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}$.)
Where Should You Go Next After Cos 315 Degrees?
Cos 315 Degrees is one point on the unit circle, and several natural doors open from here.
Trigonometric ratios of specific angles. See how every standard angle, from $30^\circ$ to $360^\circ$, gets its exact value the same way.
Unit circle with tangent. Watch sine, cosine, and tangent change together as the angle sweeps through all four quadrants.
What is a radian. Get comfortable with the radian form $\frac{7\pi}{4}$ so both units feel natural.
If your child is building these foundations, a live Bhanzu trainer teaches special-angle values starting from the unit circle and the quadrant signs in the Bhanzu trigonometry program.
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