What Is The Value Of Sin 7pi/4?
Sin 7pi/4 is $-\frac{\sqrt{2}}{2}$, equal to $-\frac{1}{\sqrt{2}}$ and approximately $-0.7071$ to four decimal places. In radians the angle is $\frac{7\pi}{4}$, and in degrees it is $315^\circ$, since $\frac{7\pi}{4} = 7 \times 45^\circ = 315^\circ$.
$$\sin\frac{7\pi}{4} = -\frac{\sqrt{2}}{2} = -\frac{1}{\sqrt{2}} \approx -0.7071$$
The value is negative for one reason: $315^\circ$ sits in the fourth quadrant, and sine is negative there. The size of the number, $\frac{\sqrt{2}}{2}$, comes straight from the $45^\circ$ special angle. For the radian side of this idea, trigonometric ratios in radians shows how the same values carry across from degree form.
How Do You Find Sin 7pi/4 Using The Reference Angle?
To find sin 7pi/4, reduce the angle to its reference angle, read the value there, then fix the sign from the quadrant. Two short steps settle it.
Step 1: find the reference angle. The reference angle is the acute angle to the nearest part of the horizontal axis. For $315^\circ$, that is $360^\circ - 315^\circ = 45^\circ$, or $\frac{\pi}{4}$ in radians. The reference value is $\sin 45^\circ = \frac{\sqrt{2}}{2}$.
Step 2: fix the sign from the quadrant. The angle $315^\circ$ lies in Quadrant IV (between $270^\circ$ and $360^\circ$). The rule ASTC names which functions stay positive by quadrant: All in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4. Sine is not on the Q4 list, so it is negative here.
Put the two together:
$$\sin\frac{7\pi}{4} = -\sin\frac{\pi}{4} = -\frac{\sqrt{2}}{2} \approx -0.7071$$
Table: Sign of sine in each quadrant, by ASTC.
Quadrant | Angle range (degrees) | Angle range (radians) | Sign of sine |
|---|---|---|---|
I | $0^\circ$ to $90^\circ$ | $0$ to $\frac{\pi}{2}$ | positive |
II | $90^\circ$ to $180^\circ$ | $\frac{\pi}{2}$ to $\pi$ | positive |
III | $180^\circ$ to $270^\circ$ | $\pi$ to $\frac{3\pi}{2}$ | negative |
IV | $270^\circ$ to $360^\circ$ | $\frac{3\pi}{2}$ to $2\pi$ | negative |
The reference-angle method works for any angle, not only this one. If reading a value in radians still feels new, what is a radian builds the idea from the ground up.
Where Does 7pi/4 Sit On The Unit Circle?
On the unit circle, the angle $\frac{7\pi}{4}$ points to the coordinate $\left(\frac{\sqrt{2}}{2},, -\frac{\sqrt{2}}{2}\right)$, and the sine of the angle is the $y$-coordinate of that point. That $y$-coordinate is $-\frac{\sqrt{2}}{2}$, which is exactly the value.
$$\text{point at } \tfrac{7\pi}{4} = \left(\cos\tfrac{7\pi}{4},; \sin\tfrac{7\pi}{4}\right) = \left(\tfrac{\sqrt{2}}{2},; -\tfrac{\sqrt{2}}{2}\right)$$
The point sits below the horizontal axis, so its height is negative. That is the whole reason sine comes out below zero at this angle, seen as a picture rather than a rule.
For a version of this diagram that also carries the tangent line, unit circle with tangent extends the same picture.
How Do You Derive Sin 7pi/4 Exactly?
The exact value also drops out of the angle-sum and angle-difference identities, with no diagram needed. Writing $\frac{7\pi}{4}$ as a special angle away from a full turn is the cleanest route.
Write $\frac{7\pi}{4}$ as $2\pi - \frac{\pi}{4}$, then apply $\sin(2\pi - x) = \sin 2\pi \cos x - \cos 2\pi \sin x$:
$$\sin\left(2\pi - \frac{\pi}{4}\right) = (0)\cos\frac{\pi}{4} - (1)\sin\frac{\pi}{4}$$
$$= -\sin\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$
A second route uses $\frac{7\pi}{4} = \frac{3\pi}{2} + \frac{\pi}{4}$ with $\sin\left(\frac{3\pi}{2} + x\right) = -\cos x$:
$$\sin\left(\frac{3\pi}{2} + \frac{\pi}{4}\right) = -\cos\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$
Both routes agree, and both agree with the unit circle. There is also a one-line shortcut: because $\frac{7\pi}{4}$ is coterminal with $-\frac{\pi}{4}$ (they differ by a full turn, $2\pi$), and sine is an odd function, $\sin\frac{7\pi}{4} = \sin\left(-\frac{\pi}{4}\right) = -\sin\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$.
Tie it back to the right triangle to close the loop. In a $45$–$45$–$90$ triangle the two legs are equal, so the ratio $\frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$. That triangle gives the size; the fourth quadrant gives the minus sign. For the triangle ratios on their own, see sin cos tan.
What Are The Related Sine Values Around 7pi/4?
All four angles built from the $\frac{\pi}{4}$ reference share the same size, $\frac{\sqrt{2}}{2}$, and differ only in sign by quadrant. Reading them together makes the sign pattern obvious.
Table: The $\frac{\pi}{4}$ family of sine values across the four quadrants.
Angle (radians) | Degrees | Quadrant | $\sin$ (exact) | $\sin$ (decimal) |
|---|---|---|---|---|
$45^\circ$ | I | $\frac{\sqrt{2}}{2}$ | $0.7071$ | |
$135^\circ$ | II | $\frac{\sqrt{2}}{2}$ | $0.7071$ | |
$225^\circ$ | III | $-\frac{\sqrt{2}}{2}$ | $-0.7071$ | |
$\frac{7\pi}{4}$ | $315^\circ$ | IV | $-\frac{\sqrt{2}}{2}$ | $-0.7071$ |
The three trigonometric ratios at $\frac{7\pi}{4}$ round out the picture. Sine and cosine share the same magnitude here, so the tangent is exactly $-1$.
Table: Sine, cosine, and tangent at $\frac{7\pi}{4}$.
Ratio | Exact value | Decimal |
|---|---|---|
$\sin\frac{7\pi}{4}$ | $-\frac{\sqrt{2}}{2}$ | $-0.7071$ |
$\frac{\sqrt{2}}{2}$ | $0.7071$ | |
$\tan\frac{7\pi}{4}$ | $-1$ | $-1.0000$ |
A fuller grid of these values across the standard angles lives in the trigonometric table.
Why Is Sin 7pi/4 Negative?
Sin 7pi/4 is negative because the angle ends in the fourth quadrant, where every point on the unit circle sits below the horizontal axis. Sine reads that height, and a height below the axis is a negative number.
The height is what sine measures. On the unit circle, sine is the $y$-coordinate of the terminal point, nothing more. At $315^\circ$ the point is $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$, so the height, and the sine, is $-\frac{\sqrt{2}}{2}$.
The fourth quadrant is below the line. Any angle between $270^\circ$ and $360^\circ$ lands in the lower-right region, below the $x$-axis. Cosine (the horizontal coordinate) stays positive there, but sine goes negative.
The size never changed. The reference angle is still $45^\circ$, so the magnitude is still $\frac{\sqrt{2}}{2}$. Only the direction, up versus down, flipped.
Cosine and sine trade roles by quadrant, which is why $\cos\frac{7\pi}{4}$ stays positive while $\sin\frac{7\pi}{4}$ turns negative at the very same angle.
Who Discovered The Sine Function?
Sine did not begin in Europe, and it did not begin as a function on a circle. It began as a table of half-chords in ancient India, and the word we use today is the result of a translation slip that stuck.
Two other figures shaped the same idea:
Hipparchus of Nicaea (c. 190–120 BCE, Greece) built the first known table of chords, the ancestor of the sine table, to predict the positions of the sun and moon.
Madhava of Sangamagrama (c. 1340–1425, India) found the infinite series for sine roughly two centuries before similar work in Europe, the same series that lets a modern calculator compute a value like this one. His work is recorded in the MacTutor History of Mathematics archive.
Where Is Sin 7pi/4 Used In The Real World?
Angles near a full turn, and the negative sine that comes with the fourth quadrant, show up wherever something rotates or oscillates.
Alternating current: household electricity rises and falls as a sine wave, and the part of each cycle past three-quarters of a turn carries a negative value, exactly the sign of sine here.
Sound and waves: a vibrating string or a loudspeaker cone moves back through its rest point on every cycle, and the downward half of the motion is where the sine of the phase angle is negative.
Navigation and GPS: bearings and positions are resolved into horizontal and vertical components using sine and cosine, and a heading into the lower-right quadrant produces a negative vertical component.
Computer graphics: rotating a sprite or a 3D model past $270^\circ$ uses the sine of the rotation angle, and the engine relies on that value being correctly negative to place the object.
One value, $-\frac{\sqrt{2}}{2}$, quietly rides inside power grids, audio, maps, and game engines. The sign is not a technicality; it tells the machine which way is down.
What Are The Most Common Mistakes With Sin 7pi/4?
Three errors account for most wrong answers on this angle, matching the confusions that surface in reference-angle help threads and worked-answer forums.
Making the answer positive.
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 the quadrant.
Don't do this:
Do not stop at the reference value. The magnitude is right, but the sign is missing.
The correct way:
Check the quadrant every time. $\frac{7\pi}{4}$ is in Quadrant IV, where sine is negative, so the answer is $-\frac{\sqrt{2}}{2}$, not $\frac{\sqrt{2}}{2}$.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types $\sin(7\pi/4)$ with the calculator set to degrees, so the device reads the input as $7\pi/4 \approx 5.5^\circ$ and returns about $0.096$.
Don't do this:
Do not enter a radian angle while the calculator is in DEGREE mode.
The correct way:
Set the calculator to RADIAN mode for $\frac{7\pi}{4}$, or convert to $315^\circ$ first and stay in DEGREE mode. Either way the display should read about $-0.7071$.
Taking the reference angle as 315°.
Where it slips in:
A student treats the whole angle, $315^\circ$, as the reference angle and looks up $\sin 315^\circ$ as if it were a first-quadrant value.
Don't do this:
Do not skip the subtraction from $360^\circ$. The reference angle is the gap to the horizontal axis, not the angle itself.
The correct way:
Compute $360^\circ - 315^\circ = 45^\circ$. The reference angle is $45^\circ$ ($\frac{\pi}{4}$), and the sine of that, with the Quadrant IV sign, gives $-\frac{\sqrt{2}}{2}$.
Practice Problems On Sin 7pi/4
Work each one, then check the answer beside it.
Convert $\frac{7\pi}{4}$ to degrees.
(Answer: $315^\circ$.)State the reference angle of $\frac{7\pi}{4}$.
(Answer: $\frac{\pi}{4}$, or $45^\circ$.)Evaluate $\cos\frac{7\pi}{4}$.
(Answer: $\frac{\sqrt{2}}{2} \approx 0.7071$, positive because cosine is positive in Quadrant IV.)Evaluate $\tan\frac{7\pi}{4}$.
(Answer: $\dfrac{\sin}{\cos} = \dfrac{-\sqrt{2}/2}{\sqrt{2}/2} = -1$.)Is $\sin\frac{7\pi}{4}$ positive or negative, and why?
(Answer: negative, because $\frac{7\pi}{4}$ is in Quadrant IV where sine is below the axis.)Find $\sin\frac{7\pi}{4} + \sin\frac{\pi}{4}$.
(Answer: $-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0$.)
Where Should You Go Next After Sin 7pi/4?
The fastest way to make this value stick is to place it next to its neighbours and its partner ratio.
Sin pi/4. The positive first-quadrant value that sets the magnitude for the whole family, including this one.
Cos 7pi/4. The partner ratio at the same angle, positive where the sine is negative, so the contrast is clear.
Sin 5pi/4. The third-quadrant sibling that shares the value $-\frac{\sqrt{2}}{2}$, useful for seeing the sign pattern twice.
If your child is learning to read these values from the unit circle rather than memorising them, a live Bhanzu trainer teaches the reference-angle method in the Bhanzu trigonometry program. It appears in India's NCERT Class 11 (Chapter 3, Trigonometric Functions) and in the United States under the Common Core high-school standards (CCSS.HSF.TF), so the same method serves students in both systems.
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