What Does Sin 45 Degrees Mean?
Sine is one of the three trigonometric ratios, in a right triangle, the sine of an angle is the side opposite it divided by the hypotenuse. So $\sin 45^\circ$ asks: in a right triangle with a $45^\circ$ angle, what fraction of the hypotenuse is the opposite side?
On the unit circle, a circle of radius $1$ centred at the origin, sine is the $y$-coordinate of the point where the angle's radius meets the circle. At $45^\circ$ that point is $\left(\dfrac{\sqrt{2}}{2},\ \dfrac{\sqrt{2}}{2}\right)$, so the $y$-coordinate, and the sine, is $\dfrac{\sqrt{2}}{2}$.
Because the point sits on the line $y = x$, its two coordinates are equal, which is the geometric reason $\sin 45^\circ = \cos 45^\circ$.
Where Does Sin 45 Degrees Show Up?
A $45^\circ$ launch angle gives a projectile its maximum range on level ground, and both its horizontal and vertical components scale with $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$, equal, which is exactly why $45^\circ$ is the sweet spot. The same value sets the diagonal of a square: a square of side $1$ has a diagonal of $\sqrt{2}$, and the diagonal makes a $45^\circ$ angle with each side.
Screen and camera diagonals, isometric drawings, and banked $45^\circ$ turns all lean on this number. It is the most-used non-trivial sine after $30^\circ$ and $60^\circ$.
Standard-Angle Reference Table
Forty-five degrees is one of the handful of angles whose sine has a clean exact form. Here are the standard first-quadrant angles in both degrees and radians.
Angle (degrees) | Angle (radians) | $\sin\theta$ (exact) | $\sin\theta$ (decimal) |
|---|---|---|---|
$0^\circ$ | $0$ | $0$ | $0.0000$ |
$30^\circ$ | $\dfrac{\pi}{6}$ | $\dfrac{1}{2}$ | $0.5000$ |
$45^\circ$ | $\dfrac{\pi}{4}$ | $\dfrac{\sqrt{2}}{2}$ | $0.7071$ |
$60^\circ$ | $\dfrac{\pi}{3}$ | $\dfrac{\sqrt{3}}{2}$ | $0.8660$ |
$90^\circ$ | $\dfrac{\pi}{2}$ | $1$ | $1.0000$ |
Read the column top to bottom and sine climbs from $0$ up to $1$ as the angle opens. At $45^\circ$ it passes through $\dfrac{\sqrt{2}}{2}$, the exact midpoint of the circle's quarter-turn, and the only standard angle where $\sin\theta = \cos\theta$.
How Do You Find The Exact Value Of Sin 45 Degrees?
Two clean routes both give $\dfrac{\sqrt{2}}{2}$.
Method 1: The 45-45-90 triangle.
Take an isosceles right triangle with both legs equal to $1$. Its two base angles are each $45^\circ$, and by the Pythagorean theorem the hypotenuse is
$$\sqrt{1^2 + 1^2} = \sqrt{2}$$
Now apply the definition, with the side opposite the $45^\circ$ angle equal to $1$:
$$\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$
The last step rationalises the denominator by multiplying top and bottom by $\sqrt{2}$.
Method 2: The unit circle.
Rotate the radius $45^\circ$ above the positive $x$-axis. Because $45^\circ$ splits the first quadrant evenly, the point lands on the line $y = x$ at $\left(\dfrac{\sqrt{2}}{2},\ \dfrac{\sqrt{2}}{2}\right)$.
$$\sin 45^\circ = y\text{-coordinate} = \frac{\sqrt{2}}{2}$$
The radian name for this angle is $\dfrac{\pi}{4}$, so $\sin\dfrac{\pi}{4} = \sin 45^\circ = \dfrac{\sqrt{2}}{2}$. The radian-first walkthrough lives at sin π/4.
Examples Of Sin 45 Degrees
Example 1
Evaluate $8\sin 45^\circ$.
$$8\sin 45^\circ = 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2} \approx 5.657$$
Example 2
Find $\sin 45^\circ$ given that $\sin 45^\circ = \cos\theta$ for an acute angle $\theta$. A student writes $\theta = 45^\circ$ and stops, then a second student says "sine and cosine are always different, so $\theta$ can't be $45^\circ$."
Wrong attempt. The second student assumes sine and cosine can never coincide.
That breaks at exactly this angle: $45^\circ$ is the one place where $\sin\theta = \cos\theta$, both equal to $\dfrac{\sqrt{2}}{2}$. So $\theta = 45^\circ$ is right after all.
Correct. By the cofunction rule $\cos\theta = \sin(90^\circ - \theta)$, and $\sin 45^\circ = \cos 45^\circ$, so $\theta = 45^\circ$. The first student had it; the "always different" reasoning was the false step.
Example 3
A right triangle has a hypotenuse of $10$ cm and a $45^\circ$ angle. Find the side opposite the $45^\circ$ angle.
$$\sin 45^\circ = \frac{\text{opposite}}{10} \implies \text{opposite} = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \approx 7.07 \text{ cm}$$
Example 4
Verify $\sin^2 45^\circ + \cos^2 45^\circ = 1$.
$$\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{1}{2} + \frac{1}{2} = 1$$
The Pythagorean identity holds, and here both terms are equal because sine and cosine match at $45^\circ$.
Example 5
A ball is thrown at $45^\circ$ with speed $20$ m/s. Find the vertical component of its velocity.
$$v_y = 20 \times \sin 45^\circ = 20 \times \frac{\sqrt{2}}{2} = 10\sqrt{2} \approx 14.14 \text{ m/s}$$
The horizontal component uses $\cos 45^\circ$ and comes out identical, which is why a $45^\circ$ throw balances height and distance.
Where Students Trip Up On Sin 45 Degrees
Mistake 1: Reporting 1/√2 when the standard form is asked
Where it slips in: Reading the value straight off the triangle ratio and stopping at $\dfrac{1}{\sqrt{2}}$.
Don't do this: Writing $\sin 45^\circ = \dfrac{1}{\sqrt{2}}$ on a problem that asks for the standard rationalised form.
The correct way: Rationalise: $\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$. Both are the same number, but $\dfrac{\sqrt{2}}{2}$ is the conventional exam form.
Mistake 2: Mixing up sin 45° with sin 30° or sin 60°
Where it slips in: Attaching the wrong radical to the wrong angle under time pressure.
Don't do this: Writing $\sin 45^\circ = \dfrac{\sqrt{3}}{2}$. That is $\sin 60^\circ$.
The correct way: Anchor on the pattern $\dfrac{\sqrt{1}}{2}, \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2}$ for $\sin 30^\circ, \sin 45^\circ, \sin 60^\circ$. The memorizer who stores the three radicals in order rarely swaps them.
Mistake 3: Forgetting the calculator's angle mode
Where it slips in: A calculator in radian mode returns $\sin(45) \approx 0.851$ instead of $0.7071$.
Don't do this: Trusting the screen without checking whether it is set to degrees.
The correct way: Confirm degree mode before entering $\sin(45)$; a reading that is not near $0.71$ signals a mode error.
Key Takeaways
Sin 45 degrees equals $\dfrac{\sqrt{2}}{2}$, about $0.7071$, an exact value because $45^\circ$ is a standard angle.
The $45$-$45$-$90$ triangle gives it as $\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$; the unit circle gives it as the $y$-coordinate on the line $y = x$.
It is the only first-quadrant angle where $\sin\theta = \cos\theta$.
The common slips are leaving the answer as $\dfrac{1}{\sqrt{2}}$ when the rationalised form is asked, and swapping it with $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$.
To drill the special angles with a teacher, explore Bhanzu's trigonometry tutor or its online math classes. The reference-angle cousin of this value in Quadrant II is sin 3π/4.
Practice These Before Moving On
Evaluate $2\sin 45^\circ + \cos 45^\circ$.
A square has side $6$ cm. Use $\sin 45^\circ$ to find the length of its diagonal.
Show that $\sin 45^\circ \times \cos 45^\circ = \dfrac{1}{2}$.
Want a live Bhanzu trainer to walk through more special-angle problems? Book a free demo class, online with an expert, anywhere.
For the geometry behind the value, see the reference on the special right triangle.
Read More
45 degree angle, the angle itself, from a geometry view.
Unit circle, where every special-angle sine is read.
Trigonometric table, sine, cosine, tangent at the standard angles.
Trigonometric ratios of specific angles, the full special-angle set.
Sin cos tan, the three ratios and how they connect.
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