Sin 315 Degrees: Exact Value, Unit Circle And Examples

What Does Sin 315 Degrees Mean?

Sin 315 degrees is the sine function evaluated at an angle of $315^\circ$. For an acute angle in a right triangle, sine is the opposite side over the hypotenuse; past $90^\circ$ no single right triangle holds the angle, so the definition moves to the unit circle.

On the unit circle, the circle of radius $1$ centred at the origin, the point at angle $\theta$ is $(\cos\theta, \sin\theta)$. Sine is the y-coordinate, so $\sin 315^\circ$ is the height of the point reached after turning $315^\circ$ counterclockwise.

Two terms carry the method:

• Quadrant. The axes divide the plane into four quadrants, numbered counterclockwise from the upper right. Angles from $270^\circ$ to $360^\circ$ lie in Quadrant IV, the lower right.

• Reference angle. The reference angle is the acute angle between the terminal side and the x-axis. For a Quadrant IV angle it is $360^\circ - \theta$, so for $315^\circ$ it is $45^\circ$.

What Is Sin 315 In Radians?

Multiply by $\frac{\pi}{180^\circ}$, the standard step in converting degrees to radians.

$$315^\circ \times \frac{\pi}{180^\circ} = \frac{7\pi}{4} \approx 5.4978 \text{ rad}$$

So $\sin\frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$ as well. This page leads with degrees; $\sin\frac{7\pi}{4}$ is the same value written for radian work.

Standard-Angle Sine Reference Table

Sine takes the same handful of sizes around the circle, with signs set by the quadrant. The row for $315^\circ$ sits among its neighbours.

The four angles with reference angle $45^\circ$ ($45^\circ$, $135^\circ$, $225^\circ$, $315^\circ$) all have sine of size $\frac{\sqrt{2}}{2}$. Only the sign changes, positive in the top half of the circle and negative in the bottom half.

Where Sin 315 Degrees Shows Up: Which Problems Use It?

Sin 315 degrees shows up whenever something points $45^\circ$ below the positive x-axis. Three places:

• Vectors and forces. A force of $10$ N at $315^\circ$ has a vertical component $10\sin 315^\circ = -5\sqrt{2} \approx -7.07$ N, pointing down. Its horizontal component is the same size, pointing right, which is what "southeast" means.

• Complex numbers. The number $1 - i$ has argument $\frac{7\pi}{4}$ (or $-\frac{\pi}{4}$), and its imaginary part divided by its modulus is $\frac{-1}{\sqrt{2}} = \sin 315^\circ$.

• Rotations. Turning a shape $315^\circ$ counterclockwise lands it in the same place as turning it $45^\circ$ clockwise, and the rotation uses $\sin 315^\circ = -\frac{\sqrt{2}}{2}$.

Why Is Sin 315 Negative?

Sin 315 is negative because the point at $315^\circ$ lies below the x-axis, and sine is the y-coordinate. Every angle between $180^\circ$ and $360^\circ$ has a negative sine.

The ASTC rule ("All Students Take Calculus") summarises the signs counterclockwise from Quadrant I:

• Quadrant I: all six functions positive

• Quadrant II: sine and cosecant positive

• Quadrant III: tangent and cotangent positive

• Quadrant IV: cosine and secant positive

In Quadrant IV only cosine and secant are positive. So $\cos 315^\circ = +\frac{\sqrt{2}}{2}$ while $\sin 315^\circ = -\frac{\sqrt{2}}{2}$, same size and opposite signs.

How Do You Find Sin 315 Degrees?

You find sin 315 degrees by taking the reference angle $45^\circ$, using $\sin 45^\circ = \frac{\sqrt{2}}{2}$, and attaching the Quadrant IV sign, which is negative for sine. The other methods confirm the same value from different starting points.

Method 1: Reference Angle And The ASTC Sign Rule

1. Quadrant: $270^\circ < 315^\circ < 360^\circ$, so Quadrant IV.

2. Reference angle: $360^\circ - 315^\circ = 45^\circ$.

3. Size: sin 45 degrees is $\frac{\sqrt{2}}{2}$.

4. Sign: sine is negative in Quadrant IV.

$$\sin 315^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}$$

Method 2: The 45-45-90 Triangle

Where does $\frac{\sqrt{2}}{2}$ come from in the first place? Take an isosceles right triangle with legs $1$ and $1$; its hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$.

$$\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$

Place that triangle below the x-axis in Quadrant IV, and its vertical leg points down, giving $-\frac{\sqrt{2}}{2}$.

Method 3: Unit Circle Coordinates

Turn a radius of length $1$ through $315^\circ$. It stops at $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$.

$$\sin 315^\circ = y = -\frac{\sqrt{2}}{2}$$

Method 4: The Angle Sum Formula

Write $315^\circ = 270^\circ + 45^\circ$ and use the sin(A + B) formula, $\sin(A + B) = \sin A\cos B + \cos A\sin B$.

$$\sin(270^\circ + 45^\circ) = \sin 270^\circ \cos 45^\circ + \cos 270^\circ \sin 45^\circ$$

$$= (-1)\left(\frac{\sqrt{2}}{2}\right) + (0)\left(\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{2}$$

Actually, the split $360^\circ - 45^\circ$ is quicker, through $\sin(360^\circ - \theta) = -\sin\theta$. The $270^\circ$ route is shown because it needs no extra identity beyond the sum formula.

Method 5: A Negative Coterminal Angle

Subtracting a full turn gives one of the coterminal angles: $315^\circ - 360^\circ = -45^\circ$. Sine is an odd function, $\sin(-\theta) = -\sin\theta$, so the sign comes out exactly once:

$$\sin 315^\circ = \sin(-45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2}$$

Method 6: Calculator Check

In degree mode, $\sin(315)$ returns $-0.7071067\ldots$. Squaring gives $0.5$, the square of $\frac{\sqrt{2}}{2}$, which confirms the exact form.

Is Sin 315 Positive Or Negative On A Calculator?

Sin 315 degrees is negative, $-0.7071$. A calculator that shows $+0.7451$ is in radian mode and has computed the sine of $315$ radians, a completely different angle.

When this page was researched, search suggestions for "what is sin 315" displayed "= 0.745133265", the radian answer. The same trap runs the other way: typing $\frac{7\pi}{4}$ in degree mode returns about $0.0958$.

How Is Sin 315 Degrees Written In Terms Of Other Trig Functions?

Sin 315 degrees can be rewritten through each of the other functions, choosing the sign for Quadrant IV. Each line equals $-\frac{\sqrt{2}}{2}$.

The minus sign in front of the square root is a choice made by the quadrant. In Quadrant III the same root would also need it; in Quadrants I and II it would not.

The cosine partner is cos 7pi/4, which is $+\frac{\sqrt{2}}{2}$, so the point is $\left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$.

What Does The Sine Graph Show At 315 Degrees?

The graph of $y = \sin\theta$ passes through $(315^\circ, -0.7071)$, on the rising stretch between the minimum $-1$ at $270^\circ$ and the zero at $360^\circ$. Nearby values climb from $\sin 300^\circ \approx -0.8660$ to $\sin 330^\circ = -0.5$.

Examples Of Sin 315 Degrees

Example 1

Evaluate $4\sin 315^\circ$.

$$4\sin 315^\circ = 4 \times \left(-\frac{\sqrt{2}}{2}\right)$$

$$= -2\sqrt{2} \approx -2.8284$$

Example 2

Find $\sin 315^\circ$ by splitting it as $270^\circ + 45^\circ$.

Wrong attempt. A student treats sine as if it spreads over a sum:

$$\sin(270^\circ + 45^\circ) = \sin 270^\circ + \sin 45^\circ = -1 + \frac{\sqrt{2}}{2} \approx -0.2929$$

Why it breaks. The unit-circle point at $315^\circ$ has y-coordinate $-0.7071$, not $-0.2929$. Sine of a sum is not the sum of sines.

Correct. Use the angle sum formula.

$$\sin 270^\circ \cos 45^\circ + \cos 270^\circ \sin 45^\circ$$

$$= (-1)\left(\frac{\sqrt{2}}{2}\right) + 0 = -\frac{\sqrt{2}}{2}$$

Example 3

Verify that $\sin^2 315^\circ + \cos^2 315^\circ = 1$.

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

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

Example 4

Evaluate $2\sin 157.5^\circ \cos 157.5^\circ$.

The sin 2x formula says $2\sin A\cos A = \sin 2A$.

$$2\sin 157.5^\circ \cos 157.5^\circ = \sin 315^\circ$$

$$= -\frac{\sqrt{2}}{2} \approx -0.7071$$

Example 5

Find $\sin(-315^\circ)$.

Sine is odd, so the minus sign comes out of the bracket.

$$\sin(-315^\circ) = -\sin 315^\circ = -\left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}$$

Check: $-315^\circ$ is coterminal with $45^\circ$ in Quadrant I, where sine is positive.

The double negative here is the step most students lose: the instinct is to see one minus sign and stop, leaving the answer negative.

Example 6

A cable pulls on a post with a force of $10$ N at $315^\circ$ from the positive x-axis. Find the vertical and horizontal components.

$$F_y = 10\sin 315^\circ = 10 \times \left(-\frac{\sqrt{2}}{2}\right) = -5\sqrt{2} \approx -7.07 \text{ N}$$

$$F_x = 10\cos 315^\circ = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \approx 7.07 \text{ N}$$

The cable pulls equally right and down, which matches a direction $45^\circ$ below the horizontal.

Where Do Students Trip Up On Sin 315 Degrees?

Mistake 1: Dropping The Sign After Finding The Reference Angle

Where it slips in: Reference-angle questions where $45^\circ$ is spotted quickly and the quadrant is never written down.

Don't do this: Writing $\sin 315^\circ = \frac{\sqrt{2}}{2}$.

The correct way: The first instinct after finding $45^\circ$ is to copy $\sin 45^\circ$ straight across, and the rusher stops there. Write the quadrant first, then the sign, then the value: Quadrant IV, negative, $-\frac{\sqrt{2}}{2}$.

Mistake 2: Swapping Sin 315 And Cos 315

Where it slips in: Questions that ask for both values, since they share the size $\frac{\sqrt{2}}{2}$.

Don't do this: Giving $\sin 315^\circ = \frac{\sqrt{2}}{2}$ and $\cos 315^\circ = -\frac{\sqrt{2}}{2}$.

The correct way: Sine is the y-coordinate, cosine is the x-coordinate. At $315^\circ$ the point is right of the y-axis and below the x-axis, so $\cos 315^\circ$ is positive and $\sin 315^\circ$ is negative.

Mistake 3: Calculator In Radian Mode

Where it slips in: Quick calculator checks after a radian question.

Don't do this: Accepting $0.7451$, the sine of $315$ radians.

The correct way: Switch to degree mode, or convert to $\frac{7\pi}{4}$ and stay in radians. A positive sine for an angle below the x-axis is the warning sign; the second-guesser who notices the mismatch but trusts the screen anyway is the one this mistake catches.

Small sign slips matter outside the classroom. In a force diagram, a vertical component of $+7.07$ N instead of $-7.07$ N reverses the direction of the pull, and every quantity computed from it inherits the error.

Where Can You Get Help With Trigonometry Values Like Sin 315 Degrees?

Values such as sin 315 degrees are where the reference-angle habit either sticks or slips. A trigonometry tutor can drill the quadrant-then-sign routine, and a high school math tutor can connect it to vectors and rotations.

For steady coverage of the unit circle, math classes online work through all four quadrants in order. Targeted math tutoring suits students who only need the sign rules fixed.

Key Takeaways

• Sin 315 degrees equals exactly $-\frac{\sqrt{2}}{2}$, about $-0.7071$, and the radian form is $\sin\frac{7\pi}{4}$.

• The angle is in Quadrant IV with reference angle $45^\circ$, and sine is negative there because the point lies below the x-axis.

• The reference-angle rule, the 45-45-90 triangle, the unit circle, the angle sum formula and the coterminal angle $-45^\circ$ all give the same value.

• At $315^\circ$ cosine is $+\frac{\sqrt{2}}{2}$ and sine is $-\frac{\sqrt{2}}{2}$; a calculator reading of $0.7451$ means radian mode.

Practice These Before Moving On

1. Find $\sin 675^\circ$.

2. Evaluate $\sin 315^\circ + \cos 315^\circ$.

3. Evaluate $\sin 315^\circ \times \cos 315^\circ$.

4. Find $\sin\frac{7\pi}{4}$ without converting to degrees.

5. Find every angle $\theta$ with $0^\circ \leq \theta < 360^\circ$ and $\sin\theta = -\frac{\sqrt{2}}{2}$.

Answers:

1. $-\frac{\sqrt{2}}{2}$, because $675^\circ - 360^\circ = 315^\circ$.

2. $0$.

3. $-\frac{1}{2}$.

4. $-\frac{\sqrt{2}}{2}$, since $\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}$ has reference angle $\frac{\pi}{4}$ in Quadrant IV.

5. $\theta = 225^\circ$ and $\theta = 315^\circ$.

If Question 2 did not come out as $0$, recheck which coordinate is sine with Mistake 2.

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