Sin 5pi: Exact Value, Unit Circle, Graph And Examples

What Does Sin 5pi Mean On The Unit Circle?

The unit circle is a circle of radius $1$ centred at the origin, and the sine of an angle is the $y$-coordinate of the point where that angle's arm meets the circle. So sin 5pi asks one question: after turning $5\pi$ radians counterclockwise from the positive $x$-axis, how high above the $x$-axis are you?

A radian is the angle that cuts off an arc equal in length to the radius, so one full turn is $2\pi$ radians ($360^\circ$). That makes $5\pi$ two full turns ($4\pi$) plus a half turn ($\pi$), see what is a radian for the arc picture.

Three words do the work here, so here they are before they are used:

• Quadrant: one of the four regions the axes cut the plane into, numbered I to IV counterclockwise from the top right. An angle whose arm lies on an axis (like $\pi$ or $5\pi$) is called quadrantal and sits in no quadrant.

• Coterminal angles: angles that end on the same arm, differing by a whole number of full turns ($2\pi k$). $5\pi$ and $\pi$ are coterminal.

• Reference angle: the acute angle between the arm and the $x$-axis. For $5\pi$ the arm lies on the axis, so the reference angle is $0$.

Standard-Angle Sine Reference Table

The table lists sine at the standard first-quadrant angles and at every half-multiple of $\pi$ up to $5\pi$. Read down the last rows and the pattern $0, 1, 0, -1$ repeats every $2\pi$.

Every whole-number multiple of $\pi$ in the table gives $0$. The neighbours sin pi and sin 3pi sit on the same point of the circle as $5\pi$, so they share its value.

Where Does Sin 5pi Show Up, And Why Does It Matter?

Sin 5pi shows up wherever a sine wave crosses zero for the fifth time after starting. A guitar string fixed at both ends vibrating in its fifth harmonic has the shape $y = A\sin\left(\frac{5\pi x}{L}\right)$, and at the far end $x = L$ the displacement is $A\sin 5\pi = 0$, which is exactly why that end stays still.

Mains electricity follows $V = V_0 \sin(2\pi f t)$, and the voltage passes through zero whenever $2\pi f t$ reaches a multiple of $\pi$. On a $50$ Hz supply, $2\pi f t = 5\pi$ happens at $t = 0.05$ s; on a $60$ Hz supply it happens at about $0.0417$ s.

What Is Sin 5pi In Degrees?

In degrees, sin 5pi is $\sin 900^\circ$, and it still equals $0$. The conversion uses $\pi$ radians $= 180^\circ$:

$$5\pi = 5 \times 180^\circ = 900^\circ$$

$$900^\circ - 2 \times 360^\circ = 180^\circ$$

So $\sin 900^\circ = \sin 180^\circ = 0$. The radians to degrees conversion is the same multiplication by $\frac{180^\circ}{\pi}$ for any angle.

How Do You Evaluate Sin 5pi?

You evaluate sin 5pi by removing full turns until the angle is between $0$ and $2\pi$, then reading the $y$-coordinate. Six methods reach the same $0$, and seeing several of them is what makes the answer stick.

Method 1: Coterminal Angle (Periodicity)

Sine repeats every $2\pi$, so $\sin(\theta + 2\pi k) = \sin\theta$ for any whole number $k$. You could subtract $2\pi$ twice; actually, it is cleaner to strip $4\pi$ in one move, because $4\pi$ is exactly two full turns.

$$\sin 5\pi = \sin(4\pi + \pi)$$

$$= \sin\pi$$

$$= 0$$

This is the idea behind any periodic function: the output depends only on where you land, not on how many laps you ran.

Method 2: Counting Half-Turns On The Unit Circle

Each $\pi$ is a half-turn, and half-turns alternate between $(1, 0)$ and $(-1, 0)$:

• $0$ ends at $(1, 0)$

• $\pi$ ends at $(-1, 0)$

• $2\pi$ ends at $(1, 0)$

• $3\pi$ ends at $(-1, 0)$

• $4\pi$ ends at $(1, 0)$

• $5\pi$ ends at $(-1, 0)$

Both points lie on the $x$-axis, so the $y$-coordinate, and the sine, is $0$ at every stop.

Method 3: The Angle Sum Identity

The identity $\sin(A + B) = \sin A\cos B + \cos A\sin B$ handles $5\pi = 4\pi + \pi$ directly. With $\sin 4\pi = 0$, $\cos 4\pi = 1$, $\sin\pi = 0$ and $\cos\pi = -1$:

$$\sin 5\pi = \sin 4\pi\cos\pi + \cos 4\pi\sin\pi$$

$$= (0)(-1) + (1)(0)$$

$$= 0$$

The full family of these rules is in sum and difference identities.

Method 4: The Double-Angle Identity

Write $5\pi = 2 \times \frac{5\pi}{2}$ and use $\sin 2A = 2\sin A\cos A$. Since $\frac{5\pi}{2}$ is coterminal with $\frac{\pi}{2}$, $\sin\frac{5\pi}{2} = 1$ and $\cos\frac{5\pi}{2} = 0$:

$$\sin 5\pi = 2\sin\frac{5\pi}{2}\cos\frac{5\pi}{2}$$

$$= 2 \times 1 \times 0$$

$$= 0$$

Method 5: Reading The Graph

On the graph of $y = \sin x$, the curve crosses the $x$-axis at $0, \pi, 2\pi, 3\pi, \ldots$ The fifth crossing after the origin is at $x = 5\pi \approx 15.708$, so the height there is $0$.

Method 6: Calculator And Series Check

In radian mode, a calculator returns $\sin(5\pi) = 0$, though a computer language may print a tiny number such as $6.12 \times 10^{-16}$ instead. That leftover comes from storing $\pi$ to about 16 significant digits, not from the mathematics; the exact value is $0$.

The calculator itself relies on the Taylor series $\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots$ after first reducing the angle by multiples of $2\pi$, which is Method 1 in disguise.

Why Is Sin Of Any Whole-Number Multiple Of Pi Equal To 0?

The rule is $\sin n\pi = 0$ for every integer $n$, and sin 5pi is the case $n = 5$. Every multiple of $\pi$ is a whole number of half-turns, and every half-turn stops on the $x$-axis at either $(1, 0)$ or $(-1, 0)$.

The general solution of $\sin x = 0$ is therefore $x = n\pi$, a fact used constantly in trigonometric equations. Cosine behaves differently at the same angles: $\cos n\pi = (-1)^n$, which is why cos pi is $-1$ while sine is $0$.

What Does The Graph Of Sin X Look Like Near 5pi?

Near $5\pi$ the sine curve is falling as it crosses the axis, because $5\pi$ is an odd multiple of $\pi$. Just after the crossing the curve dips below zero, and just before it the curve is above zero.

For a small step $h$, $\sin(5\pi + h) = -\sin h \approx -h$. With $h = 0.1$, $\sin(5\pi + 0.1) \approx -0.0998$, and the approximation $-0.1$ is off by less than $0.0002$; the error grows quickly once $h$ passes about $0.25$ ($15^\circ$).

Examples Of Sin 5pi

Example 1

Evaluate $3\sin 5\pi + 2\cos 5\pi$.

Use $\sin 5\pi = 0$ and $\cos 5\pi = -1$.

$$3(0) + 2(-1)$$

$$= -2$$

Final answer: $-2$

Example 2

A student says that because $5$ is odd, $\sin 5\pi = -1$. Check the claim.

The tempting route copies the pattern from cosine: odd multiples of $\pi$ give $-1$, so $\sin 5\pi = -1$.

Test it against Method 2. The angle $5\pi$ ends at $(-1, 0)$, and the sine is the $y$-coordinate of that point, which is $0$, not $-1$.

The $-1$ is the $x$-coordinate, so the student read the wrong coordinate.

Final answer: $\sin 5\pi = 0$; it is $\cos 5\pi$ that equals $-1$.

Example 3

Find $\sin\left(5\pi + \frac{\pi}{6}\right)$.

Use $\sin(5\pi + \theta) = -\sin\theta$, which follows from Method 3 with $\cos 5\pi = -1$.

$$\sin\left(5\pi + \frac{\pi}{6}\right) = -\sin\frac{\pi}{6}$$

$$= -\frac{1}{2}$$

Final answer: $-\frac{1}{2}$

Example 4

Find all solutions of $\sin x = 0$ with $0 \leq x \leq 6\pi$.

The solutions are the multiples of $\pi$ in the interval:

$$x = 0, \pi, 2\pi, 3\pi, 4\pi, 5\pi, 6\pi$$

Final answer: seven solutions, including $x = 5\pi$.

The habit that saves marks here is listing the endpoints: readers who count only the interior crossings stop at five answers and lose $0$ and $6\pi$.

Example 5

A point moves on a circle of radius $2$ m, starting at $(2, 0)$ and turning counterclockwise through $5\pi$ radians. How far is it above the $x$-axis, and how far has it travelled?

Height $= r\sin\theta$:

$$2\sin 5\pi = 2 \times 0 = 0 \text{ m}$$

Distance travelled is the arc length $s = r\theta$:

$$s = 2 \times 5\pi = 10\pi \approx 31.42 \text{ m}$$

Final answer: height $0$ m, distance about $31.42$ m.

Example 6

The displacement of a string in its fifth harmonic is $y = 4\sin\left(\frac{5\pi x}{L}\right)$ mm. Find $y$ at $x = L$ and at $x = \frac{L}{10}$.

At $x = L$:

$$y = 4\sin 5\pi = 0 \text{ mm}$$

At $x = \frac{L}{10}$:

$$y = 4\sin\frac{\pi}{2} = 4 \text{ mm}$$

Final answer: $0$ mm at the clamped end and $4$ mm at the first antinode.

Where Do Students Trip Up On Sin 5pi?

Mistake 1: Using The Cosine Pattern For Sine

Where it slips in: Right after learning that $\cos n\pi = (-1)^n$, when a radian angle with an odd multiple of $\pi$ appears.

Don't do this: Write $\sin 5\pi = -1$ because $5$ is odd.

The correct way: Sine is the $y$-coordinate, and every multiple of $\pi$ lands on the $x$-axis, so $\sin 5\pi = 0$. Students who have just met the cosine alternation almost always carry it over to sine on the next problem; saying "sine is height" out loud before answering breaks the habit.

Mistake 2: Leaving The Calculator In Degree Mode

Where it slips in: Typing $\sin(5\pi)$ on a calculator set to degrees.

Don't do this: Accept the output $0.2707$, which is the sine of $5\pi \approx 15.708$ degrees.

The correct way: Switch to radian mode, or convert first: $5\pi = 900^\circ$ and $\sin 900^\circ = 0$. The rusher who never checks the mode is the reader most surprised by this one.

Mistake 3: Treating A Tiny Computer Output As Non-Zero

Where it slips in: Spreadsheets and programming languages print $\sin(5\pi)$ as about $6.12 \times 10^{-16}$.

Don't do this: Report that sin 5pi is "slightly positive" or use the tiny number in a later division.

The correct way: Treat any result that small as rounding noise from storing $\pi$ approximately, and use the exact value $0$. The second-guesser who trusts the exact reasoning first, then checks the screen, gets this right.

Where Can You Get Help With Trigonometry Values?

Values like sin 5pi click fastest when someone walks you around the unit circle and checks your half-turn counting in real time. A trigonometry tutor can drill radian reduction with you, and a high school math tutor can connect it to graphs and equations.

For a regular weekly slot, math classes online cover the full trigonometry sequence, and math tutoring works well for one tricky chapter.

Key Takeaways

• Sin 5pi equals exactly $0$, because $5\pi$ ends at $(-1, 0)$ on the unit circle.

• $5\pi = 4\pi + \pi$, so $5\pi$ is coterminal with $\pi$; in degrees it is $900^\circ$.

• The rule $\sin n\pi = 0$ holds for every integer $n$, while $\cos n\pi = (-1)^n$.

• The angle sum, double-angle, graph and calculator routes all confirm the same value.

• Watch for the cosine pattern, degree mode, and tiny floating-point outputs.

Practice These Before Moving On

1. Evaluate $\sin 5\pi + \sin\frac{5\pi}{2} + \cos 5\pi$.

2. Find $\sin\left(5\pi - \frac{\pi}{4}\right)$.

3. How many solutions does $\sin x = 0$ have for $0 < x < 5\pi$?

4. Convert $5\pi$ to degrees and give $\sin$ of the result.

Answers:

1. $0 + 1 + (-1) = 0$

2. $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$

3. Four ($\pi, 2\pi, 3\pi, 4\pi$; both endpoints are excluded)

4. $900^\circ$, and $\sin 900^\circ = 0$

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Read More

• Sin 2pi and a full turn

• Sine function, graph and properties

• Coterminal angles explained

• The unit circle

• Cos 3pi value