Sin 150 Degrees : Exact Value 1/2 and How to Find It

What Does Sin 150 Degrees Mean?

Sine is one of the three trigonometric ratios. In a right triangle it is the opposite side over the hypotenuse, but $150^\circ$ is obtuse, so no right triangle holds it. The unit circle fills that gap.

On the unit circle (radius $1$, centred at the origin), the sine of an angle is the y-coordinate of the point the radius reaches. Rotating $150^\circ$ counterclockwise from the positive x-axis lands the radius at $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. The y-coordinate is $\frac{1}{2}$, so $\sin 150^\circ = \frac{1}{2}$. The x-coordinate is negative and the y-coordinate is positive — the signature of the second quadrant for sin, cos, and tan.

Where Does Sin 150 Degrees Show Up?

A regular dodecagon - a twelve-sided polygon - has interior angles of exactly $150^\circ$, which is why $150^\circ$ turns up in tiling and clock-face geometry (at $5$ o'clock the hands sit $150^\circ$ apart). In physics, a projectile launched at $150^\circ$ from a reference direction has the same vertical component as one at $30^\circ$, since both share $\sin 150^\circ = \sin 30^\circ$. Any obtuse triangle solved with the applications of trigonometry can carry a $150^\circ$ angle, and its sine sets the triangle's height.

Standard-Angle Reference Table

A quadrant is one of the four quarters the axes cut the plane into, counted counterclockwise from the top-right. The angle $150^\circ$ opens past $90^\circ$ but stays under $180^\circ$, so it sits in the second quadrant, where sine (the y-coordinate) is positive. Here are the standard angles from $0^\circ$ through $180^\circ$ in degrees and radians.

The sine column climbs to $1$ at $90^\circ$, then mirrors back down. That symmetry pairs $150^\circ$ with $30^\circ$ and $120^\circ$ with $60^\circ$, since supplementary angles carry the same sine. The complete set of values sits in the trigonometric table.

How Do You Find The Exact Value Of Sin 150 Degrees?

The reference angle - the acute angle between the terminal side and the x-axis - gives the size, and the quadrant gives the sign.

Method 1: The reference angle.

For an angle between $90^\circ$ and $180^\circ$, the reference angle is $180^\circ - \theta$.

$$180^\circ - 150^\circ = 30^\circ$$

Sine is positive in the second quadrant, so:

$$\sin 150^\circ = +\sin 30^\circ = \frac{1}{2}$$

Method 2: The supplementary-angle identity.

The identity $\sin(180^\circ - \theta) = \sin\theta$ pairs an angle with its supplement. Writing $150^\circ = 180^\circ - 30^\circ$:

$$\sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}$$

The third-quadrant partner $210^\circ$ shares the same $30^\circ$ reference angle but has a negative sine, $-\frac{1}{2}$; you can see the sign flip on the sin 210 degrees page.

Method 3: The decimal check.

A calculator in degree mode returns $\sin(150) = 0.5$ exactly, matching the fraction $\frac{1}{2}$ with no rounding, because $\frac{1}{2}$ is rational.

Examples Of Sin 150 Degrees

Example 1

Evaluate $8\sin 150^\circ$.

$$8\sin 150^\circ = 8 \times \frac{1}{2} = 4$$

Example 2

Find $\sin 150^\circ$ from its reference angle.

Wrong attempt. A student reasons that $150^\circ$ is close to $180^\circ$, where sine is $0$, and sine is close to $1$ at $90^\circ$, so guesses $\sin 150^\circ = \frac{\sqrt{3}}{2}$.

Check it: the table shows $\sin 150^\circ = 0.5$, and $\frac{\sqrt{3}}{2} \approx 0.87$ is the value of $\sin 120^\circ$, not $\sin 150^\circ$. Guessing by "closeness" mixes the two obtuse angles up.

Correct. Use the reference angle: $180^\circ - 150^\circ = 30^\circ$, so $\sin 150^\circ = \sin 30^\circ = \frac{1}{2}$. The value shrinks toward $0$ as the angle heads to $180^\circ$, so $150^\circ$ giving $0.5$ is exactly right.

Example 3

A triangle has an angle of $150^\circ$ between two sides of length $6$ and $10$. Its area is $\frac{1}{2} \times 6 \times 10 \times \sin 150^\circ$. Find it.

$$\text{Area} = \frac{1}{2} \times 6 \times 10 \times \frac{1}{2} = 15 \text{ square units}$$

Example 4

Verify that $\sin 150^\circ = \sin 30^\circ$ using the supplementary identity.

$$\sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}$$

Both sides equal $\frac{1}{2}$, so the identity checks out.

Example 5

Express $\sin 150^\circ$ in radians and evaluate $\sin\left(\frac{5\pi}{6}\right)$.

Since $150^\circ = \frac{5\pi}{6}$ radians, $\sin\left(\frac{5\pi}{6}\right) = \sin 150^\circ = \frac{1}{2}$. The two forms name one angle and one value.

Where Students Trip Up On Sin 150 Degrees

Mistake 1: Confusing sin 150° with sin 120°

Where it slips in: Both are obtuse second-quadrant angles, and their values ($\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$) get swapped under time pressure.

Don't do this: Writing $\sin 150^\circ = \frac{\sqrt{3}}{2}$.

The correct way: The bigger the second-quadrant angle, the smaller the sine. $\sin 150^\circ = \frac{1}{2}$ (reference $30^\circ$); $\sin 120^\circ = \frac{\sqrt{3}}{2}$ (reference $60^\circ$). The habit that fixes this is always computing the reference angle first.

Mistake 2: Making the sine negative

Where it slips in: Assuming a second-quadrant angle always produces a negative ratio.

Don't do this: Writing $\sin 150^\circ = -\frac{1}{2}$.

The correct way: In the second quadrant the y-coordinate is positive, so sine is positive: $\sin 150^\circ = +\frac{1}{2}$. Cosine and tangent are the negative ones there.

Mistake 3: Reading the x-coordinate instead of the y-coordinate

Where it slips in: Grabbing the wrong number off the point $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Don't do this: Writing $\sin 150^\circ = -\frac{\sqrt{3}}{2}$, which is actually $\cos 150^\circ$.

The correct way: Sine is the y-coordinate, $\frac{1}{2}$; cosine is the x-coordinate, $-\frac{\sqrt{3}}{2}$. A negative $150^\circ$ "sine" almost always means cosine got read by mistake.

Key Takeaways

• Sin 150 degrees equals $\frac{1}{2}$, exactly $0.5$, an exact value built on the $30^\circ$ reference angle.

• The angle is in the second quadrant, where sine stays positive; the reference angle $30^\circ$ sets the size.

• In radians, $\sin 150^\circ = \sin\left(\frac{5\pi}{6}\right)$, and by the supplementary identity $\sin 150^\circ = \sin 30^\circ$.

• The common slips are swapping it with $\sin 120^\circ$ and adding a stray negative sign.

• To go further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or math classes online.

Practice These Before Moving On

1. Evaluate $\sin 150^\circ + \cos 150^\circ$ using exact values.

2. A triangle has sides $9$ and $4$ with a $150^\circ$ angle between them; find its area using $\sin 150^\circ$.

3. Show that $\sin 150^\circ = \cos 60^\circ$ and explain the cofunction reason.

Want a live Bhanzu trainer to walk through more sin 150 degrees problems? Book a free demo class.

Read More

• Sin 120 degrees — the other second-quadrant standard angle, equal to $\frac{\sqrt{3}}{2}$.

• Sin 180 degrees — where the sine curve returns to zero.

• Cofunction identities — why $\sin 150^\circ = \cos 60^\circ$.

• Pythagorean identities — the identity linking sine and cosine at any angle.

• Sum and difference formulas — build exact values like $\sin 150^\circ$ from known angles.