Sin Pi/2 : Exact Value on the Unit Circle (= 1)

What Does Sin Pi/2 Mean?

Sine is one of the three trigonometric ratios, and its cleanest definition lives on the unit circle - a circle of radius $1$ centred at the origin. For any angle, the sine is the $y$-coordinate of the point where the angle's radius meets the circle.

Sweep the radius through $\frac{\pi}{2}$ radians and it points straight up, meeting the circle at $(0, 1)$. The $y$-coordinate there is $1$, so sin pi/2 is $1$.

Where Does Sin Pi/2 Show Up?

Radian angles are the native language of waves and rotation, so sin pi/2 marks the crest in almost every oscillation written in radians. In the wave $y = \sin t$, the first peak of $1$ arrives at $t = \frac{\pi}{2}$, which is why $\frac{\pi}{2}$ is called a "quarter-phase" shift.

It also fixes the link between sine and cosine: shifting a cosine curve right by $\frac{\pi}{2}$ turns it into a sine curve, because $\cos\left(t - \frac{\pi}{2}\right) = \sin t$. That single quarter-turn is the reason the two functions are the same shape offset in time.

What Is The Standard-Angle Sine Reference Table?

Here are the first-quadrant angles in radian order, with $\frac{\pi}{2}$ closing the climb. Sine grows from $0$ to $1$ as the arc opens from $0$ to $\frac{\pi}{2}$.

The bottom row is sin pi/2. Whatever notation the angle wears, the height at a quarter-turn is $1$.

What Is A Radian, And Why Does Pi/2 Sit At The Top?

Before reading the value, it helps to know what the angle $\frac{\pi}{2}$ measures. A radian is the angle that wraps an arc equal in length to the radius; a full circle is $2\pi$ radians, so a quarter of that turn is $\frac{2\pi}{4} = \frac{\pi}{2}$.

That quarter-turn lands the radius pointing straight up. Because sine tracks vertical height on the circle, an angle that points straight up gives the largest possible height, which is why sin pi/2 is the peak.

How Do You Find The Exact Value Of Sin Pi/2?

Two clean routes reach $1$: the unit circle, and the radian-to-degree bridge.

Method 1: The unit circle.

Rotate the radius counter-clockwise by $\frac{\pi}{2}$ radians from the positive $x$-axis.

It stops at the top of the circle, the point $(0, 1)$.

$$\sin\left(\frac{\pi}{2}\right) = y\text{-coordinate} = 1$$

Method 2: Convert to degrees first.

If radians feel abstract, convert using $\pi \text{ rad} = 180^\circ$, so $\frac{\pi}{2} = 90^\circ$. Then read the known value $\sin 90^\circ = 1$, the same answer the degree twin sin 90 degrees reaches by starting from the degree measure.

Method 3: The calculator check.

Put the calculator in radian mode and enter $\sin(\pi \div 2)$, which returns $1$. Entering $\sin(90)$ in radian mode instead gives $0.8939$, because that reads the angle as $90$ radians.

Examples Of Sin Pi/2

Example 1

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

$$4\sin\left(\frac{\pi}{2}\right) = 4 \times 1 = 4$$

Example 2

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

Wrong attempt. A student assumes both terms are "special-angle" values equal to $1$ and writes $1 + 1 = 2$.

Check $\sin(\pi)$ on the unit circle. Half a turn lands at $(-1, 0)$, whose $y$-coordinate is $0$, so $\sin(\pi) = 0$, not $1$.

Correct. Using $\sin\left(\frac{\pi}{2}\right) = 1$ and $\sin(\pi) = 0$:

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

See the neighbouring value in value of sin pi.

Example 3

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

At $\frac{\pi}{2}$ the point is $(0, 1)$, so $\cos\left(\frac{\pi}{2}\right) = 0$.

$$(1)^2 + (0)^2 = 1 + 0 = 1$$

The Pythagorean identity checks out.

Example 4

Solve $\sin t = 1$ for $0 \le t \le 2\pi$.

Sine reaches its maximum of $1$ once in a full turn.

$$t = \frac{\pi}{2}$$

Example 5

A wave is modelled by $y = \sin t$. At what time does it first reach its peak, and what is that peak?

The peak is the maximum of sine, which is $1$, and it first occurs at $t = \frac{\pi}{2}$.

$$y = \sin\left(\frac{\pi}{2}\right) = 1$$

Where Students Trip Up On Sin Pi/2

Mistake 1: Reading pi/2 as an ordinary number, not an angle

Where it slips in: Treating $\frac{\pi}{2} \approx 1.57$ as if $1.57$ itself were the sine value.

Don't do this: Writing $\sin\left(\frac{\pi}{2}\right) \approx 1.57$.

The correct way: $\frac{\pi}{2}$ is the angle you feed in, not the answer. Feed the angle $\frac{\pi}{2}$ (a quarter-turn) into sine and the output is $1$. Students meeting radians for the first time often confuse the input angle with the output value.

Mistake 2: Leaving the calculator in degree mode for a radian angle

Where it slips in: Entering $\sin(\pi \div 2)$ while the calculator is set to degrees.

Don't do this: Trusting a screen reading of $0.0274$, which is $\sin$ of $1.57^\circ$.

The correct way: Switch to radian mode before entering a radian angle. A near-zero answer for sin pi/2 is the tell that the mode is wrong.

Mistake 3: Confusing sin pi/2 with sin pi

Where it slips in: Blurring the two half-related angles $\frac{\pi}{2}$ and $\pi$.

Don't do this: Writing $\sin\left(\frac{\pi}{2}\right) = 0$, which is actually $\sin(\pi)$.

The correct way: $\frac{\pi}{2}$ is a quarter-turn (top of the circle, height $1$); $\pi$ is a half-turn (left of the circle, height $0$). Sketching the point before reading the height stops the mix-up.

Key Takeaways

• Sin pi/2 equals $1$, read as the height of the quarter-turn point $(0, 1)$ on the unit circle.

• The angle $\frac{\pi}{2}$ is a quarter of a full turn of $2\pi$ radians, which is $90^\circ$.

• In degrees, $\sin\left(\frac{\pi}{2}\right) = \sin 90^\circ = 1$, the same value in a different unit.

• The common slip is reading $\frac{\pi}{2} \approx 1.57$ as the answer; it is the input angle, and the output is $1$.

To build radian fluency 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 $2\sin\left(\frac{\pi}{2}\right) + 3\cos\left(\frac{\pi}{2}\right)$.

2. Convert $\frac{\pi}{2}$ to degrees, then state $\sin$ of that angle.

3. Find every $t$ in $0 \le t \le 2\pi$ with $\sin t = 1$.

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

Read More

• Applications of trigonometry — where sine values do real work.

• The full trigonometric table — every standard angle in degrees and radians.

• Radians to degrees — the conversion behind $\frac{\pi}{2} = 90^\circ$.

• Cofunction identities — the $\frac{\pi}{2}$ shift that links sine and cosine.

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