Sin 105 Degrees: Exact Value & Formula

What Is The Value Of Sin 105 Degrees?

The value of Sin 105 Degrees is $\dfrac{\sqrt{6}+\sqrt{2}}{4}$, which equals approximately $0.9659$ (to four decimal places, $0.9659$). This is an exact value, a surd, not a rounded decimal.

The angle can be written two ways, and instructional work should always show both:

• Degrees: $105^\circ$

• Radians: $\dfrac{7\pi}{12}$, because $105 \times \dfrac{\pi}{180} = \dfrac{7\pi}{12}$.

So in one line:

$$\sin 105^\circ = \sin\frac{7\pi}{12} = \frac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659$$

Because $105^\circ$ is more than $90^\circ$ but less than $180^\circ$, it lands in the second quadrant, where sine is positive. That is why the answer carries a plus sign, a point we prove properly further down. For the special-angle values this rests on, see the trigonometric table.

How Do You Find The Exact Value Of Sin 105 Degrees?

$105^\circ$ is not one of the memorised angles, but it is a sum of two that are: $60^\circ$ and $45^\circ$. That lets us use the angle-sum identity for sine.

The identity is:

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$

Set $A = 60^\circ$ and $B = 45^\circ$. For the underlying rule and more worked cases, see sin(A + B) and the full set of sum and difference identities.

Now substitute the known values of sin 60 degrees and sin 45 degrees, working one line at a time:

$$\sin 105^\circ = \sin(60^\circ + 45^\circ)$$

$$= \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ$$

$$= \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} + \frac{1}{2}\cdot\frac{\sqrt{2}}{2}$$

$$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$$

$$= \frac{\sqrt{6}+\sqrt{2}}{4}$$

Final answer: $\sin 105^\circ = \dfrac{\sqrt{6}+\sqrt{2}}{4} \approx 0.9659$.

A quick numerical check confirms it: $\sqrt{6}\approx 2.4495$ and $\sqrt{2}\approx 1.4142$, so the top is about $3.8637$, and dividing by $4$ gives $0.9659$. That matches what a calculator returns for $\sin 105^\circ$.

Can You Check Sin 105 Degrees A Second Way?

Yes, and checking your own answer is a habit worth keeping. Split the angle differently: $105^\circ = 150^\circ - 45^\circ$, then use the angle-difference identity $\sin(A - B) = \sin A \cos B - \cos A \sin B$.

$$\sin 105^\circ = \sin(150^\circ - 45^\circ)$$

$$= \sin 150^\circ \cos 45^\circ - \cos 150^\circ \sin 45^\circ$$

$$= \frac{1}{2}\cdot\frac{\sqrt{2}}{2} - \left(-\frac{\sqrt{3}}{2}\right)\cdot\frac{\sqrt{2}}{2}$$

$$= \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4}$$

Two different splits, one answer. That is the mark of a correct exact value.

Where Does 105° Sit On The Unit Circle?

On the unit circle, an angle is measured anticlockwise from the positive $x$-axis, and the point where the angle's arm meets the circle has coordinates $(\cos\theta, \sin\theta)$. The sine is simply the $y$-coordinate, the height of that point above the horizontal axis.

At $105^\circ$ the arm has swung past the top of the circle into the second quadrant, up and to the left. Its coordinates are:

$$(\cos 105^\circ,\ \sin 105^\circ) = \left(-\tfrac{\sqrt{6}-\sqrt{2}}{4},\ \tfrac{\sqrt{6}+\sqrt{2}}{4}\right) \approx (-0.2588,\ 0.9659)$$

The $x$-coordinate is negative (the point is left of centre) but the $y$-coordinate is positive and high, close to the top of the circle. That height is $\sin 105^\circ$, and its nearness to $1$ tells you the value should be close to $1$, exactly as $0.9659$ is.

This also ties $\sin 105^\circ$ back to a right triangle. Drop a vertical line from the point on the circle to the $x$-axis, and you get a right triangle whose hypotenuse is the radius $1$ and whose height is $\sin 105^\circ$. The height over the hypotenuse, opposite over hypotenuse, is the same value the addition formula gave.

How Do You Use The Reference Angle And Quadrant For Sin 105 Degrees?

For any angle past $90^\circ$, two questions settle the value: what is the reference angle, and what sign does the quadrant give?

The reference angle is the acute angle between the terminal arm and the $x$-axis. In the second quadrant you measure from the negative $x$-axis, so:

$$\text{reference angle} = 180^\circ - 105^\circ = 75^\circ$$

The sign comes from the ASTC rule (All, Sine, Tangent, Cosine positive as you move through quadrants one to four). In Quadrant II, only sine and its reciprocal are positive. So:

$$\sin 105^\circ = +\sin 75^\circ = \sin 75^\circ$$

That is a useful shortcut on its own: $\sin 105^\circ$ and $\sin 75^\circ$ are the same number, because $105^\circ$ and $75^\circ$ are supplementary ($105^\circ + 75^\circ = 180^\circ$), and supplementary angles share the same sine. You can confirm the value against sin 75 degrees, which also equals $\dfrac{\sqrt{6}+\sqrt{2}}{4}$.

How Does Sin 105 Degrees Relate To Cos 15 Degrees?

There is a second identity hiding in this angle. The cofunction relationship says the sine of an angle equals the cosine of its complement:

$$\sin\theta = \cos(90^\circ - \theta)$$

A cleaner route for an obtuse angle uses $\sin\theta = \cos(\theta - 90^\circ)$:

$$\sin 105^\circ = \cos(105^\circ - 90^\circ) = \cos 15^\circ$$

So $\sin 105^\circ = \cos 15^\circ = \dfrac{\sqrt{6}+\sqrt{2}}{4}$. You can verify this directly against cos 15 degrees. For the general rule, see the cofunction identities.

Table: Sin 105 Degrees inside its family of related special-angle values.

Notice how the table reads across: $\sin 105^\circ = \sin 75^\circ$ (supplementary), and $\sin 105^\circ = \cos 15^\circ$ (cofunction). The same surd keeps appearing because these angles are all built from $15^\circ$ steps. For the radian forms in one place, see trigonometric ratios in radians.

Why Is Sin 105 Degrees Positive?

The plus sign is not a convention, it follows from where the angle points. Three ways of seeing it, all giving the same conclusion:

• From the unit circle: sine is the $y$-coordinate. At $105^\circ$ the point sits above the $x$-axis, so its height is positive.

• From ASTC: the second quadrant is the "S" quadrant, where sine is positive and cosine is negative. That is exactly what the table shows, $\sin 105^\circ > 0$ and $\cos 105^\circ < 0$.

• From the supplementary link: $\sin 105^\circ = \sin 75^\circ$, and $75^\circ$ is a first-quadrant acute angle whose sine is plainly positive.

The value is also close to $1$ for a reason: $90^\circ$ has the maximum sine of $1$, and $105^\circ$ is only $15^\circ$ past the top, so it has barely started coming down. That is the intuition behind $0.9659$ being so near the ceiling. The sine function rises to $1$ at $90^\circ$ and eases back down through the second quadrant.

Who Discovered The Angle-Sum Formula Behind Sin 105 Degrees?

The trick that cracks $\sin 105^\circ$, breaking an awkward angle into a sum of familiar ones, is almost two thousand years old. It began with a table of chords, not sines.

Two other figures shaped the same idea:

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

• Aryabhata (476–550 CE, India) tabulated the half-chord, the quantity he called jya, which travelled through Arabic into Latin and became our word "sine." His tables made angle values a computational tool, not just a geometric curiosity.

Where Is Sin 105 Degrees Used In The Real World?

Values like $\sin 105^\circ$ are not classroom-only. Any time an angle past a right angle meets a wave, a rotation, or a measurement, this kind of sine appears.

• Waves and alternating current: an AC voltage is a sine wave, and its value at a phase of $105^\circ$ is $\sin 105^\circ$ of the peak, so engineers reading a waveform at that phase are reading this exact height.

• Surveying and navigation: the law of sines solves triangles with obtuse angles, and a bearing or a plot boundary of $105^\circ$ feeds its sine straight into the calculation of an unknown distance.

• Structural engineering: a strut or roof member set at $105^\circ$ to a beam carries a load whose vertical component depends on $\sin 105^\circ$, which decides how much weight the joint really takes.

• Computer graphics and animation: rotating a point by $105^\circ$ uses $\sin 105^\circ$ and $\cos 105^\circ$ inside the rotation formula, placing the point exactly where the turn should land.

• Physics of projectiles and optics: launch angles and angles of incidence beyond $90^\circ$ (measured from a chosen axis) resolve into components using the sine of the obtuse angle.

One surd, $\frac{\sqrt{6}+\sqrt{2}}{4}$, quietly runs through power grids, land surveys, rooftops, and screens. That reach is the reason exact trigonometric values are worth knowing, not just approximating.

What Are The Most Common Mistakes With Sin 105 Degrees?

These four errors account for most wrong answers on this angle, matching the confusions that show up in precalculus addition-formula guides and public homework threads.

Leaving the calculator in the wrong mode.

Where it slips in:

A student types $\sin 105$ expecting $0.9659$ but the calculator is set to radians, so it returns about $0.9705$, the sine of $105$ radians, and the whole problem goes off course.

Don't do this:

Do not trust a decimal without checking the angle unit first.

The correct way:

Set the calculator to degree mode for $\sin 105^\circ$, or convert to $\frac{7\pi}{12}$ and stay in radian mode. Confirm the unit before reading any value. See what is a radian.

Assuming an obtuse angle makes sine negative.

Where it slips in:

A student sees that $105^\circ$ is bigger than $90^\circ$ and writes a negative answer, confusing sine with cosine, which really is negative here.

Don't do this:

Do not attach a minus sign just because the angle is obtuse.

The correct way:

Use ASTC. In Quadrant II sine is positive, so $\sin 105^\circ = +0.9659$. It is $\cos 105^\circ$ that is negative.

Taking the reference angle as $90^\circ - 105^\circ$.

Where it slips in:

A student subtracts from $90^\circ$ out of habit and gets $-15^\circ$, then loses track of the sign and the value.

Don't do this:

Do not use $90^\circ - \theta$ for a second-quadrant angle.

The correct way:

In Quadrant II the reference angle is $180^\circ - \theta$, so it is $180^\circ - 105^\circ = 75^\circ$, and $\sin 105^\circ = \sin 75^\circ$.

Distributing sine over the sum.

Where it slips in:

A student writes $\sin(60^\circ + 45^\circ) = \sin 60^\circ + \sin 45^\circ$, which gives about $1.573$, a value sine can never reach.

Don't do this:

Do not split sine across an addition. Sine is not linear.

The correct way:

Use the full identity $\sin(A+B) = \sin A\cos B + \cos A\sin B$. The cross-terms are what keep the answer inside the range $-1$ to $1$.

Practice Problems On Sin 105 Degrees

Try each before reading the answer.

1. Write $\sin 105^\circ$ in exact surd form.
   (Answer: $\dfrac{\sqrt{6}+\sqrt{2}}{4}$.)

2. State $\sin 105^\circ$ as a decimal to four places.
   (Answer: $0.9659$.)

3. Convert $105^\circ$ to radians.
   (Answer: $\dfrac{7\pi}{12}$.)

4. Use the supplementary relationship to name another angle with the same sine.
   (Answer: $75^\circ$, since $\sin 105^\circ = \sin 75^\circ$.)

5. Which cosine equals $\sin 105^\circ$?
   (Answer: $\cos 15^\circ$, by the cofunction relationship.)

6. Find $\cos 105^\circ$ using the angle-sum formula.
   (Answer: $\cos 60^\circ\cos 45^\circ - \sin 60^\circ\sin 45^\circ = \dfrac{\sqrt{2}-\sqrt{6}}{4} \approx -0.2588$.)

Where Should You Go Next After Sin 105 Degrees?

Once one compound angle makes sense, several natural doors open.

1. Sum and difference identities. The full toolkit that turns any awkward angle into a sum or difference of familiar ones.

2. Sin 75 degrees. The supplementary partner of $105^\circ$, worked through the same way, to lock in the pattern.

3. Sin, cos, tan. Step back to the three core ratios and how they connect through the right triangle and the unit circle.

If your child is building these foundations, a live Bhanzu trainer teaches angle values starting from the "why", the unit circle and the addition formula, rather than a table to memorise, in the Bhanzu trigonometry program.