Sin 135 Degrees: Exact Value, Radians & Unit Circle

What Is The Value Of Sin 135 Degrees?

Sin 135 degrees equals $\dfrac{\sqrt{2}}{2}$, an exact value that is also written $\dfrac{1}{\sqrt{2}}$ and rounds to $0.7071$ (more precisely $0.70710678$). The angle in radians is $\dfrac{3\pi}{4}$, so the same fact is written $\sin\frac{3\pi}{4} = \dfrac{\sqrt{2}}{2}$.

$$\sin 135^\circ = \sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \approx 0.7071$$

The value is positive. That single fact separates $\sin 135^\circ$ from $\cos 135^\circ$, which equals the negative amount $-\frac{\sqrt{2}}{2}$. The rest of this page shows where the value comes from, first from the right triangle, then from the unit circle, then from two identities.

How Do You Find Sin 135 Degrees?

The fastest route uses the reference angle and the sign rule for the quadrant. Three short steps settle it.

• Place the angle. $135^\circ$ is between $90^\circ$ and $180^\circ$, so it lands in the second quadrant.

• Find the reference angle. In the second quadrant the reference angle is $180^\circ - 135^\circ = 45^\circ$.

• Fix the sign. Sine is positive in the second quadrant (the ASTC or CAST rule: in Quadrant II only Sine and its reciprocal are positive).

Putting the three together gives the value directly:

$$\sin 135^\circ = +\sin 45^\circ = \frac{\sqrt{2}}{2}$$

The reference angle tells you the size of the answer, and the quadrant tells you the sign. For a wider drill on the ASTC pattern across all four quadrants, see trigonometric ratios of specific angles.

How Does The Right Triangle Give Sin 45°?

The reference angle $45^\circ$ comes from a 45-45-90 triangle, the shape you get by cutting a square along its diagonal. Its two legs are equal, say each of length $1$, and the hypotenuse is $\sqrt{2}$ by the Pythagorean theorem.

$$\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

So the size of $\sin 135^\circ$ is a right-triangle ratio, and the second-quadrant sign keeps it positive. The same ratio appears on the full trigonometric table.

Where Does 135° 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 it lands on has coordinates $(\cos\theta, \sin\theta)$. The sine of the angle is the $y$-coordinate of that point.

At $135^\circ$, the point sits in the upper-left quarter of the circle, at:

$$\left(-\frac{\sqrt{2}}{2},; \frac{\sqrt{2}}{2}\right)$$

The $y$-coordinate is $+\frac{\sqrt{2}}{2}$, which is exactly $\sin 135^\circ$. The $x$-coordinate is negative, which is why $\cos 135^\circ$ comes out negative while $\sin 135^\circ$ stays positive.

For a version of this diagram that also carries the tangent line, see unit circle with tangent.

How Do You Derive The Exact Value Of Sin 135 Degrees?

The reference-angle method gives the answer quickly, but two identities prove the same value from first principles. Each one is worth knowing because it transfers to angles that have no neat reference-angle shortcut.

Method 1: The supplementary-angle identity.

Because $135^\circ$ and $45^\circ$ add to $180^\circ$, they are supplementary, and $\sin(180^\circ - \theta) = \sin\theta$:

$$\sin 135^\circ = \sin(180^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}$$

Method 2: The angle-sum identity.

Write $135^\circ$ as $90^\circ + 45^\circ$ and expand with $\sin(A + B) = \sin A\cos B + \cos A\sin B$:

$$\sin 135^\circ = \sin 90^\circ \cos 45^\circ + \cos 90^\circ \sin 45^\circ$$

$$= (1)\left(\frac{\sqrt{2}}{2}\right) + (0)\left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}$$

The tools for this step live at sum and difference identities.

Method 3: The cofunction relation.

Since $\sin(90^\circ + \theta) = \cos\theta$, setting $\theta = 45^\circ$ gives:

$$\sin 135^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$$

All three methods land on the same exact value, $\frac{\sqrt{2}}{2}$. More of these angle relationships are collected under cofunction identities.

How Does Sin 135° Compare To Nearby Angles?

Reading $\sin 135^\circ$ next to the other special angles makes its value easy to remember. Notice the mirror symmetry: $\sin(180^\circ - \theta) = \sin\theta$, so the sine climbs to $1$ at $90^\circ$ and then retraces the same values back down.

Table: Sine of the common special angles, in degrees and radians.

The row for $135^\circ$ matches the row for $45^\circ$, and the row for $120^\circ$ matches $60^\circ$. Each obtuse angle borrows the sine of its acute partner. The radian column follows the same pattern, which you can read in full at trigonometric ratios in radians.

Why Is Sin 135 Degrees Positive?

The sign of a sine value is decided entirely by which half of the circle the angle points into. Height above the $x$-axis is positive; height below it is negative.

• The point is above the axis. $135^\circ$ lands in the second quadrant, in the upper-left of the unit circle, so its $y$-coordinate, the sine, is positive.

• The mirror partner is acute. $135^\circ$ reflects across the vertical axis onto $45^\circ$, and both share the same height, so they share the same sine.

• Cosine tells the other half of the story. The point's $x$-coordinate is negative in the second quadrant, so $\cos 135^\circ = -\frac{\sqrt{2}}{2}$ while $\sin 135^\circ$ stays positive.

That contrast, positive sine and negative cosine, is the whole reason a single angle needs both the reference angle (for size) and the quadrant (for sign). A compact reference on the three core ratios lives at sin cos tan.

Who Discovered The Sine Values Behind Sin 135 Degrees?

Long before calculators, mathematicians built tables of these values by hand so that astronomers and navigators could look them up. The story of sine is really the story of those tables, and it stretches across the ancient Mediterranean and India.

Two more figures shaped the sine table that makes $\sin 135^\circ$ a lookup rather than a mystery:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) is often called the founder of trigonometry for compiling the first known table of chords, the ancestor of the sine table.

• Ptolemy (c. 100–170 CE, Roman Egypt) extended that work in the Almagest, tabulating chords in half-degree steps to power his model of the heavens.

Where Is Sin 135 Degrees Used In The Real World?

An obtuse angle like $135^\circ$ shows up wherever something points backward-and-upward, or where a wave has swung past its peak. The sine of such angles is not a classroom-only quantity.

• Projectile symmetry. A ball launched at $135^\circ$ has no forward range, but $\sin 135^\circ = \sin 45^\circ$ captures why a $45^\circ$ launch and its supplement share the same vertical behaviour, the basis of the classic "complementary launch angles give equal range" result.

• Alternating current. AC voltage traces $\sin\theta$ as the generator turns, so the instantaneous voltage at the $135^\circ$ point of the cycle is $\frac{\sqrt{2}}{2}$ of the peak, on the falling side after the crest.

• Navigation and bearings. A heading of $135^\circ$ is due south-east, and resolving it into north and east components uses the sine and cosine of that obtuse bearing.

• Computer graphics. Rotating a sprite or a light vector by $135^\circ$ multiplies its coordinates by $\sin 135^\circ$ and $\cos 135^\circ$, placing it in the upper-left of the screen space.

One value, $\frac{\sqrt{2}}{2}$, sits behind a thrown ball, a power socket, a compass heading, and a rotated game asset. The same special angle keeps reappearing across fields that look unrelated.

What Are The Most Common Mistakes With Sin 135 Degrees?

These four slips account for most wrong answers on second-quadrant angles, and each one has a clean fix.

Making the sine negative.

Where it slips in:

A student sees that $135^\circ$ is "past $90^\circ$" and assumes every ratio turns negative there, so they write $\sin 135^\circ = -\frac{\sqrt{2}}{2}$.

Don't do this:

Do not apply a blanket minus sign to the whole quadrant. Only cosine and tangent are negative in the second quadrant.

The correct way:

Use ASTC. In Quadrant II, Sine (and cosecant) stay positive, so $\sin 135^\circ = +\frac{\sqrt{2}}{2}$. The negative sign belongs to $\cos 135^\circ$ and $\tan 135^\circ$, not to sine.

Leaving the calculator in radian mode.

Where it slips in:

A student types $\sin(135)$ with the calculator set to radians and reads off $\approx -0.088$, then trusts the wrong number.

Don't do this:

Do not enter a degree value while the MODE is set to radians. $135$ radians is a completely different angle from $135^\circ$.

The correct way:

Set the calculator to degree mode for $\sin 135^\circ$, or convert first: $135^\circ = \frac{3\pi}{4}$ radians, then evaluate $\sin\frac{3\pi}{4}$ in radian mode. Both give $0.7071$.

Misreading the reference angle.

Where it slips in:

A student uses $135^\circ$ itself as the reference angle, or subtracts from the wrong axis and gets $135^\circ - 90^\circ = 45^\circ$ by luck without understanding why.

Don't do this:

Do not guess the reference angle. In the second quadrant it is measured from the negative $x$-axis, not from $0^\circ$ or $90^\circ$.

The correct way:

For a second-quadrant angle, the reference angle is $180^\circ - \theta$. Here $180^\circ - 135^\circ = 45^\circ$, and $\sin 135^\circ = \sin 45^\circ$.

Copying the cosine sign onto the sine.

Where it slips in:

Having just found $\cos 135^\circ = -\frac{\sqrt{2}}{2}$, a student carries the minus sign across and writes $\sin 135^\circ = -\frac{\sqrt{2}}{2}$ too.

Don't do this:

Do not assume sine and cosine share a sign. At $135^\circ$ they have the same size but opposite signs.

The correct way:

Read the coordinates separately. The point is $\left(-\frac{\sqrt{2}}{2}, +\frac{\sqrt{2}}{2}\right)$, so cosine (the $x$-value) is negative and sine (the $y$-value) is positive. Compare the two directly at cos 135 degrees.

Practice Problems On Sin 135 Degrees

Work each one, then check against the answer that follows.

1. Convert $135^\circ$ to radians.
   (Answer: $\frac{3\pi}{4}$.)

2. Evaluate $\cos 135^\circ$.
   (Answer: $-\frac{\sqrt{2}}{2} \approx -0.7071$, negative because the $x$-coordinate is negative in Quadrant II.)

3. Evaluate $\tan 135^\circ$.
   (Answer: $\tan 135^\circ = \frac{\sin 135^\circ}{\cos 135^\circ} = \frac{\sqrt{2}/2}{-\sqrt{2}/2} = -1$.)

4. Use the reference angle to find $\sin 225^\circ$.
   (Answer: $225^\circ$ is in Quadrant III with reference angle $45^\circ$ and sine negative, so $-\frac{\sqrt{2}}{2}$.)

5. Find $\csc 135^\circ$, the reciprocal of $\sin 135^\circ$.
   (Answer: $\frac{1}{\sqrt{2}/2} = \sqrt{2} \approx 1.4142$.)

6. Explain in one line why $\sin 135^\circ = \sin 45^\circ$.
   (Answer: $135^\circ$ and $45^\circ$ are supplementary, and $\sin(180^\circ - \theta) = \sin\theta$.)

Where Should You Go Next After Sin 135 Degrees?

$\sin 135^\circ$ is one entry in the wider map of special angles, and several natural doors open from here.

1. Sin 45 degrees. The acute partner whose value $135^\circ$ borrows, and the cleanest place to see the 45-45-90 triangle.

2. Sin 3pi/4. The identical fact written in radians, for when the angle arrives as $\frac{3\pi}{4}$ instead of $135^\circ$.

3. What is a radian. The unit behind $\frac{3\pi}{4}$, and why radians, not degrees, are the natural measure for trigonometric functions.

If your child is building these foundations, a live Bhanzu trainer teaches special-angle values starting from the unit circle and the right triangle, not from memorised tables, in the Bhanzu trigonometry program.