Sin Pi/4 : Exact Value √2/2 Explained

What Does Sin Pi/4 Mean?

Sine is one of the three trigonometric ratios: in a right triangle it is the side opposite the angle divided by the hypotenuse. So $\sin\frac{\pi}{4}$ asks what fraction of the hypotenuse the opposite side reaches when the angle is $45^\circ$.

On the unit circle, sine is the $y$-coordinate of the point where the angle's radius meets the circle. Sweeping through $\frac{\pi}{4}$ radians (a radian being the arc-equals-radius angle) lands the radius at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, so the height, and therefore sin pi/4, is $\frac{\sqrt{2}}{2}$.

Where Does Sin Pi/4 Show Up?

A $\frac{\pi}{4}$ tilt (that is, $45^\circ$) splits a quantity evenly between two directions, so sin pi/4 is the fraction that appears whenever something is aimed halfway. A projectile launched at $45^\circ$ splits its speed equally into horizontal and vertical parts, each scaled by $\frac{\sqrt{2}}{2}$, which is why $45^\circ$ gives the maximum range on flat ground.

The value also runs through anything built on a square's diagonal. A diagonal cuts a square into two 45-45-90 triangles, and the diagonal of a unit square has length $\sqrt{2}$, so the vertical drop across a $45^\circ$ slope scales with $\frac{\sqrt{2}}{2}$.

What Is The Standard-Angle Sine Reference Table?

The angle $\frac{\pi}{4}$ radians sits exactly in the middle of the first quadrant, so its sine sits in the middle of the climb. Reading the table in radian order, $\frac{\pi}{4}$ takes the value $\frac{\sqrt{2}}{2}$, between $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$.

The angle $\frac{\pi}{4}$ is the only standard first-quadrant angle whose sine and cosine are equal, since $\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$. That balance is what makes it the symmetric middle of the table.

How Do You Find The Exact Value Of Sin Pi/4?

Two routes reach $\frac{\sqrt{2}}{2}$: the 45-45-90 triangle, or the unit circle.

Method 1: The 45-45-90 triangle.

Take a right triangle whose two legs are each $1$. Because the legs match, the two base angles are equal at $\angle 45^\circ$ each, and by the Pythagorean theorem the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$.

Apply the sine definition to a $45^\circ$ angle, then rationalise:

$$\sin\frac{\pi}{4} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

Method 2: The unit circle.

Rotate a radius of length $1$ through $\frac{\pi}{4}$ radians.

Its tip lands at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, sitting on the line $y = x$.

$$\sin\frac{\pi}{4} = y\text{-coordinate} = \frac{\sqrt{2}}{2}$$

Method 3: The calculator check.

In radian mode, $\sin(\pi \div 4)$ returns $0.7071067\ldots$. Squaring $\frac{\sqrt{2}}{2}$ gives $\frac{2}{4} = \frac{1}{2}$, whose square root is that same $0.7071$, confirming the exact form.

Examples Of Sin Pi/4

Example 1

Evaluate $2\sin\left(\frac{\pi}{4}\right)$.

$$2\sin\left(\frac{\pi}{4}\right) = 2 \times \frac{\sqrt{2}}{2} = \sqrt{2} \approx 1.414$$

Example 2

Evaluate $\sin\left(\frac{\pi}{4}\right)$ by scaling down from $\sin\left(\frac{\pi}{2}\right)$.

Wrong attempt. A student reasons that $\frac{\pi}{4}$ is half of $\frac{\pi}{2}$, so its sine should be half of $\sin\frac{\pi}{2} = 1$, giving $\frac{1}{2}$.

Sine does not scale in a straight line with the angle. Check against the table: $\sin\frac{\pi}{4} \approx 0.71$, not $0.5$, so halving the angle did not halve the sine.

Correct. Read it from the 45-45-90 triangle instead:

$$\sin\frac{\pi}{4} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.71$$

Example 3

Verify the identity $\sin^2\left(\frac{\pi}{4}\right) + \cos^2\left(\frac{\pi}{4}\right) = 1$.

Since $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ as well:

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

The Pythagorean identity holds.

Example 4

A right triangle has a hypotenuse of $10$ cm and a $\frac{\pi}{4}$ angle. Find the side opposite that angle.

$$\sin\frac{\pi}{4} = \frac{\text{opposite}}{10} \implies \text{opposite} = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \approx 7.07 \text{ cm}$$

Example 5

Evaluate $\sin\left(\frac{5\pi}{4}\right)$ using the reference angle.

The angle $\frac{5\pi}{4}$ lies in the third quadrant, where sine is negative, and its reference angle is $\frac{\pi}{4}$.

$$\sin\frac{5\pi}{4} = -\sin\frac{\pi}{4} = -\frac{\sqrt{2}}{2}$$

The full third-quadrant working is in sin 5pi/4.

Where Students Trip Up On Sin Pi/4

Mistake 1: Leaving the answer as 1/√2 when the rationalised form is expected

Where it slips in: Reading straight off the 45-45-90 triangle and stopping at $\frac{1}{\sqrt{2}}$.

Don't do this: Submitting $\frac{1}{\sqrt{2}}$ where the exam wants a rationalised denominator.

The correct way: Multiply top and bottom by $\sqrt{2}$ to get the standard form $\frac{\sqrt{2}}{2}$. The two are equal in value, but $\frac{\sqrt{2}}{2}$ is the form most answer keys expect.

Mistake 2: Assuming sine scales linearly with the angle

Where it slips in: Trying to find $\sin\frac{\pi}{4}$ by halving $\sin\frac{\pi}{2}$.

Don't do this: Writing $\sin\frac{\pi}{4} = \frac{1}{2}$.

The correct way: Sine is a curve, not a straight line, so half the angle is not half the sine. Students first meeting the special angles often expect a proportional pattern; use the 45-45-90 triangle to get $\frac{\sqrt{2}}{2} \approx 0.71$ instead.

Mistake 3: Forgetting the quadrant sign for related angles

Where it slips in: Extending sin pi/4 to angles like $\frac{5\pi}{4}$ or $\frac{3\pi}{4}$ without checking the quadrant.

Don't do this: Writing $\sin\frac{5\pi}{4} = \frac{\sqrt{2}}{2}$.

The correct way: The reference angle gives the size $\frac{\sqrt{2}}{2}$; the quadrant supplies the sign. In the third quadrant sine is negative, so $\sin\frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$.

Key Takeaways

• Sin pi/4 equals $\frac{\sqrt{2}}{2}$, about $0.7071$, an exact value because $\frac{\pi}{4}$ is a standard angle.

• The 45-45-90 triangle gives it as $\frac{1}{\sqrt{2}}$, which rationalises to $\frac{\sqrt{2}}{2}$; the unit circle gives it as the $y$-coordinate at $\frac{\pi}{4}$.

• In degrees, $\sin\frac{\pi}{4} = \sin 45^\circ = \frac{\sqrt{2}}{2}$, and it is the only standard angle with $\sin = \cos$.

• The most common slip is halving $\sin\frac{\pi}{2}$ to get $\frac{1}{2}$: sine is a curve, not a straight line.

To take the standard angles further with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or structured math classes online.

Practice These Before Moving On

1. Evaluate $\sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right)$.

2. A square has side $6$ cm. Use $\sin\frac{\pi}{4}$ to find the vertical rise across its diagonal from one corner.

3. Evaluate $\sin\left(\frac{3\pi}{4}\right)$ using the reference angle, and state its sign.

Want a live Bhanzu trainer to walk through more standard-angle problems? Book a free demo class.

Read More

• Sin 45 degrees — the degree twin of this angle, same value $\frac{\sqrt{2}}{2}$.

• The full trigonometric table — every standard angle in one place.

• Radians to degrees — the conversion behind $\frac{\pi}{4} = 45^\circ$.

• Cofunction identities — why $\sin\frac{\pi}{4} = \cos\frac{\pi}{4}$.

• Sin, cos, and tan explained — the three ratios at a glance.