Cos pi/2 : Exact Value: 0, and the Unit Circle

What Does Cos pi/2 Mean?

On the unit circle, cosine is the $x$-coordinate of the point where the angle's radius meets the circle. Sweep the radius counterclockwise by $\dfrac{\pi}{2}$ and it points straight up, landing at $(0, 1)$, so the cosine is $0$.

The same value comes from the trigonometric ratios as an angle opens toward a quarter-turn: the adjacent side shrinks to nothing while the hypotenuse stays fixed, driving the ratio to $0$. Radians and degrees name the same quarter-turn; only the unit differs.

Where Does Cos pi/2 Show Up?

In wave and rotation problems, which almost always run on radians, $\cos\left(\dfrac{\pi}{2}\right) = 0$ marks the quarter-turn where a cosine wave crosses zero. A cosine curve starts at its peak and hits zero exactly at $\dfrac{\pi}{2}$.

The value also captures a right angle in radian form: two directions a quarter-turn apart have zero cosine between them, the signal that they are perpendicular on the unit circle.

Standard-Angle Reference Table

A radian measures an angle by arc length, and $\dfrac{\pi}{2}$ is a quarter-turn, the point where cosine reaches the bottom of its first-quadrant slide. Here is the standard set.

The degree twin of this value lives at cos 90 degrees, the same $0$ reached from the degree angle instead of the radian.

How Do You Find The Exact Value Of Cos pi/2?

At exactly $\dfrac{\pi}{2}$ the right triangle collapses, so the unit circle is the home definition.

Method 1: The unit circle.

Sweep the radius by $\dfrac{\pi}{2}$ from the positive $x$-axis and it points straight up to $(0, 1)$. Cosine is the $x$-coordinate:

$$\cos\left(\frac{\pi}{2}\right) = x\text{-coordinate of }(0, 1) = 0$$

Method 2: Convert to degrees.

Radians convert to degrees by multiplying by $\dfrac{180^\circ}{\pi}$:

$$\frac{\pi}{2} \times \frac{180^\circ}{\pi} = 90^\circ$$

Then $\cos 90^\circ = 0$. Full conversion detail lives at radians to degrees. Both routes give $0$.

Examples Of Cos pi/2

Example 1

Evaluate $6\cos\left(\dfrac{\pi}{2}\right) + 5$.

$$6\cos\left(\frac{\pi}{2}\right) + 5 = 6 \times 0 + 5 = 5$$

Example 2

A student evaluates $\cos\left(\dfrac{\pi}{2}\right)$ on a calculator set to degrees, types $\cos(\pi/2)$, and reads $\approx 1.0$. What went wrong?

Wrong attempt. The student records $\cos\left(\dfrac{\pi}{2}\right) \approx 1$ from the screen.

That cannot be right: cosine equals $1$ only at an angle of $0$, and $\dfrac{\pi}{2}$ is a full quarter-turn away. The calculator read the number $\dfrac{\pi}{2} \approx 1.571$ as $1.571$ degrees, an almost-zero angle.

Correct. Switch to radian mode, or convert first: $\dfrac{\pi}{2} = 90^\circ$, and $\cos 90^\circ = 0$.

Example 3

Simplify $\dfrac{8\cos\left(\frac{\pi}{2}\right)}{\cos 0}$.

$$\frac{8\cos\left(\frac{\pi}{2}\right)}{\cos 0} = \frac{8 \times 0}{1} = 0$$

Example 4

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

$$0^2 + 1^2 = 0 + 1 = 1$$

The Pythagorean identity holds at the quarter-turn.

Example 5

Why is $\tan\left(\dfrac{\pi}{2}\right)$ undefined, given $\cos\left(\dfrac{\pi}{2}\right) = 0$?

Tangent is $\dfrac{\sin\theta}{\cos\theta}$, so $\tan\left(\dfrac{\pi}{2}\right) = \dfrac{1}{0}$. Division by zero is undefined, which is why tangent has a vertical asymptote at $\dfrac{\pi}{2}$.

Where Students Trip Up On Cos pi/2

Mistake 1: Leaving the calculator in degree mode

Where it slips in: Typing a radian angle into a calculator still set to degrees.

Don't do this: Trusting $\cos(\pi/2) \approx 1$ from the screen.

The correct way: A radian angle needs radian mode. The student who never checks the mode meets a near-$1$ value; the expected $0$ is the tell that the setting is wrong.

Mistake 2: Reading cos pi/2 as sin pi/2

Where it slips in: Assuming both functions peak at the quarter-turn.

Don't do this: Writing $\cos\left(\dfrac{\pi}{2}\right) = 1$.

The correct way: At $\dfrac{\pi}{2}$ the point is $(0, 1)$: sine is the height $1$, cosine is the width $0$. The student who reads the coordinate as width-then-height stops swapping the two.

Mistake 3: Dividing by cos pi/2 without noticing it is zero

Where it slips in: Simplifying an expression with a hidden $\cos\left(\dfrac{\pi}{2}\right)$ in a denominator.

Don't do this: Treating $\dfrac{1}{\cos\left(\frac{\pi}{2}\right)}$ as an ordinary number.

The correct way: $\dfrac{1}{\cos\left(\frac{\pi}{2}\right)} = \dfrac{1}{0}$ is undefined; this is why $\sec\left(\dfrac{\pi}{2}\right)$ and $\tan\left(\dfrac{\pi}{2}\right)$ do not exist.

Key Takeaways

• Cos pi/2 equals $0$, exactly, read as the $x$-coordinate of the unit-circle point $(0, 1)$.

• $\dfrac{\pi}{2}$ radians is $90^\circ$, so cos pi/2 and $\cos 90^\circ$ are the same exact value.

• Because $\cos\left(\dfrac{\pi}{2}\right) = 0$, both $\tan\left(\dfrac{\pi}{2}\right)$ and $\sec\left(\dfrac{\pi}{2}\right)$ are undefined.

• The commonest radian error is degree mode on the calculator, which returns a near-$1$ value instead of $0$.

To build radian and unit-circle confidence with a teacher, explore Bhanzu's trigonometry tutor, its high school math tutor programme, or math tutoring online.

Practice These Before Moving On

1. Evaluate $3\cos\left(\dfrac{\pi}{2}\right) + 2\cos 0$.

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

3. Explain in one line why $\sec\left(\dfrac{\pi}{2}\right)$ does not exist.

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Read More

• Cos 2pi — the full-turn value, $\cos 2\pi = 1$.

• Cos pi/3 — the standard radian angle valued $\dfrac{1}{2}$.

• Sin, Cos, Tan — the three ratios defined together.

• Trigonometric Ratios in Radians — reading the standard angles in radian form.

• Reference Angle — how larger angles borrow a first-quadrant value.