Sin 7pi/4: Exact Value On The Unit Circle

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.

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.

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 $\frac{7\pi}{4}$.

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.

1. Convert $\frac{7\pi}{4}$ to degrees.
   (Answer: $315^\circ$.)

2. State the reference angle of $\frac{7\pi}{4}$.
   (Answer: $\frac{\pi}{4}$, or $45^\circ$.)

3. Evaluate $\cos\frac{7\pi}{4}$.
   (Answer: $\frac{\sqrt{2}}{2} \approx 0.7071$, positive because cosine is positive in Quadrant IV.)

4. Evaluate $\tan\frac{7\pi}{4}$.
   (Answer: $\dfrac{\sin}{\cos} = \dfrac{-\sqrt{2}/2}{\sqrt{2}/2} = -1$.)

5. 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.)

6. 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.

1. Sin pi/4. The positive first-quadrant value that sets the magnitude for the whole family, including this one.

2. Cos 7pi/4. The partner ratio at the same angle, positive where the sine is negative, so the contrast is clear.

3. 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.