Cos 75 Degrees: Exact Value And How To Find It

What Is The Value Of Cos 75 Degrees?

Cos 75 degrees equals $\dfrac{\sqrt{6} - \sqrt{2}}{4}$, and as a decimal that is $0.2588$ to four places. Because $75^\circ$ lies in the first quadrant, where every trigonometric ratio is positive, the answer is a small positive number rather than a negative one.

The same angle written in radians is $\frac{5\pi}{12}$, since $75 \times \frac{\pi}{180} = \frac{5\pi}{12} \approx 1.3090$ radians. Instructional trigonometry always carries both forms, so keep them paired:

$$\cos 75^\circ = \cos\frac{5\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588$$

That surd form is exact. The decimal is a rounded stand-in for it, useful for a quick estimate but never a substitute for the closed form in a proof.

How Do You Find Cos 75 Degrees?

The cleanest route splits $75^\circ$ into two angles you already know, $45^\circ$ and $30^\circ$, then applies the cosine sum identity. The identity is $\cos(A + B) = \cos A \cos B - \sin A \sin B$, and the minus sign in the middle is the part students most often drop.

Start by writing the split:

$$\cos 75^\circ = \cos(45^\circ + 30^\circ)$$

Apply the identity with $A = 45^\circ$ and $B = 30^\circ$:

$$\cos 75^\circ = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ$$

Substitute the standard values, $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, and $\sin 30^\circ = \frac{1}{2}$:

$$\cos 75^\circ = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$$

Multiply each product:

$$\cos 75^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$$

Combine over the common denominator:

$$\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588$$

A quick numeric check confirms it: $\sqrt{6} \approx 2.4495$ and $\sqrt{2} \approx 1.4142$, so $\frac{2.4495 - 1.4142}{4} = \frac{1.0353}{4} = 0.2588$. That matches a calculator to four places. The full sum and difference identities are collected in the sum and difference identities reference, and the building-block values live in cos 45 degrees and cos 30 degrees.

Is There A Second Way To Derive Cos 75 Degrees?

Yes. The half-angle route reaches the same answer from $\cos 150^\circ$, since $75^\circ$ is exactly half of $150^\circ$. This is a useful cross-check when you want to confirm the sum-identity result without re-using it.

The half-angle formula for cosine is:

$$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$

Set $\theta = 150^\circ$, so $\frac{\theta}{2} = 75^\circ$. Because $75^\circ$ is in the first quadrant, take the positive root. Using $\cos 150^\circ = -\frac{\sqrt{3}}{2}$:

$$\cos 75^\circ = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2}$$

Numerically, $\sqrt{2 - \sqrt{3}} = \sqrt{0.2679} \approx 0.5176$, so the result is $0.2588$, the same value. The two forms $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $\frac{\sqrt{2 - \sqrt{3}}}{2}$ are algebraically equal, they just look different on the page.

Why Does Cos 75 Degrees Equal Sin 15 Degrees?

Cos 75 degrees equals $\sin 15^\circ$ because $75^\circ$ and $15^\circ$ are complementary, they add to $90^\circ$. The cofunction relationship says $\cos\theta = \sin(90^\circ - \theta)$, so the cosine of any angle is the sine of its complement.

$$\cos 75^\circ = \sin(90^\circ - 75^\circ) = \sin 15^\circ$$

Both equal $\frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588$, which you can confirm against the sin 15 degrees page. This is not a coincidence of these two angles, it is the general cofunction identities rule at work, the same rule catalogued under trigonometric ratios of complementary angles. Its mirror image is also true here: $\sin 75^\circ = \cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$.

Where Does 75 Degrees Sit On The Unit Circle?

On the unit circle, the point for $75^\circ$ has coordinates $(\cos 75^\circ, \sin 75^\circ) = (0.2588, 0.9659)$. The cosine is the $x$-coordinate and the sine is the $y$-coordinate, so a steep angle like $75^\circ$ lands high up and close to the vertical axis, with a small $x$ and a large $y$.

The right-triangle picture agrees with the circle. Drop the vertical from the point $P$ to the $x$-axis and you form a right triangle with hypotenuse $1$ (the radius). The side adjacent to the $75^\circ$ angle has length $0.2588$, so $\cos 75^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{0.2588}{1} = 0.2588$.

Triangle and circle give one answer, which is the whole point of anchoring the value both ways. You can explore the tangent line and the other ratios on the interactive unit circle with tangent page.

What Are The Related Values Around Cos 75 Degrees?

The table below places $75^\circ$ inside the family of special first-quadrant angles, with each row in both degrees and radians. Notice how $15^\circ$ and $75^\circ$ trade their sine and cosine, the cofunction pairing from earlier.

Table: Sine, cosine, and tangent for the special angles around 75°, exact forms.

For the wider set of standard values, see trigonometric ratios of specific angles and the full trigonometric table. The neighbouring cornerstones have their own pages at cos 60 degrees and cos 15 degrees.

What Is Cos 75 Degrees In Terms Of Other Functions?

Once you have $\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}$, the sibling ratios follow directly. The sine of the same angle is $\sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$, and dividing gives the tangent:

$$\tan 75^\circ = \frac{\sin 75^\circ}{\cos 75^\circ} = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} = 2 + \sqrt{3} \approx 3.7321$$

The reciprocal is the secant, $\sec 75^\circ = \frac{1}{\cos 75^\circ} = \frac{4}{\sqrt{6} - \sqrt{2}} = \sqrt{6} + \sqrt{2} \approx 3.8637$. Each of these is exact, and each traces back to the single cosine value derived above.

Who Discovered The Identities Behind Cos 75 Degrees?

The formula that cracks $\cos 75^\circ$, the cosine of a sum, is nearly two thousand years old. Long before anyone wrote $\cos(A + B)$, astronomers needed the same relationship to predict where planets would be, and they built it out of chords in a circle rather than the sine and cosine we use today.

Two other figures shaped this same corner of trigonometry:

• Hipparchus of Nicaea (c. 190 – c. 120 BCE, Greece) is often called the founder of trigonometry, and he compiled one of the earliest known tables of chords, the ancestor of every sine table since.

• Aryabhata (476 – 550 CE, India) tabulated the sine function itself, which he called jya, in his work the Aryabhatiya, moving the subject from chords toward the half-chord that became our modern sine.

Where Is Cos 75 Degrees Used In The Real World?

A $75^\circ$ tilt is steep but common, and its cosine, the horizontal reach of that tilt, shows up wherever something is aimed sharply upward.

• Roofing and structural engineering: a steeply pitched roof or a truss brace set near $75^\circ$ uses the cosine to find the short horizontal run for a given rafter length.

• Satellite dishes and antennas: aiming a dish at a high-elevation satellite involves a steep angle, and the cosine converts the slant reach into the horizontal offset from the mount.

• Camera gimbals and robotics: a camera arm or robotic joint tipped to $75^\circ$ resolves into horizontal and vertical components using cosine and sine, which is how the controller knows where the lens is pointing.

• Optics and physics: light striking a surface at a steep angle of incidence uses the cosine of that angle in the equations for reflection and refraction.

• Navigation and astronomy: measuring a star or landmark high in the sky returns a steep elevation angle, and its cosine feeds the horizontal-distance calculation.

One small number, $0.2588$, is the horizontal signature of a steep angle across every one of these fields. The applications of trigonometry page collects more of these cases.

What Are The Most Common Mistakes With Cos 75 Degrees?

These three errors account for most wrong answers on $\cos 75^\circ$, drawn from the confusions that recur across public math forums for this exact angle.

Leaving the calculator in radian mode.

Where it slips in:

A student types cos(75) expecting degrees, but the calculator is set to radians and returns about $0.9218$, a completely different number.

Don't do this:

Do not trust a decimal that does not match $0.2588$ for this angle. A wrong mode is the single most common source of a wrong trigonometric value.

The correct way:

Set the calculator to degree mode before entering $75$, or enter the radian form $\frac{5\pi}{12}$ if the calculator is in radian mode. Confirm the result reads $0.2588$.

Using a plus sign in the cosine sum formula.

Where it slips in:

A student writes $\cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ$, copying the sine formula's sign by mistake.

Don't do this:

Do not use a plus between the two products for a cosine of a sum. That plus belongs to the cosine of a difference, not a sum.

The correct way:

Cosine of a sum takes a minus: $\cos(A + B) = \cos A \cos B - \sin A \sin B$. The sign flips: sum uses minus, difference uses plus.

Treating cos 75° as if it equalled cos 15° or sin 75°.

Where it slips in:

A student sees the surd $\frac{\sqrt{6} \pm \sqrt{2}}{4}$ and grabs the wrong sign, reporting $\frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$, which is actually $\cos 15^\circ$ and $\sin 75^\circ$.

Don't do this:

Do not swap the plus and minus forms. $\cos 75^\circ$ is the smaller value because $75^\circ$ is a steep angle with a short horizontal reach.

The correct way:

Anchor the sign with a sanity check: a steep angle has a small cosine, so $\cos 75^\circ = \frac{\sqrt{6} - \sqrt{2}}{4} \approx 0.2588$, the smaller surd. If your answer is close to $1$, you found the sine or the wrong angle.

Practice Problems On Cos 75 Degrees

Work each one, then check against the answer that follows.

1. Convert $75^\circ$ to radians.
   (Answer: $75 \times \frac{\pi}{180} = \frac{5\pi}{12}$ radians.)

2. Use $75^\circ = 45^\circ + 30^\circ$ to write the exact value of $\cos 75^\circ$.
   (Answer: $\frac{\sqrt{6} - \sqrt{2}}{4}$.)

3. Evaluate $\cos 75^\circ$ as a decimal to four places.
   (Answer: $0.2588$.)

4. Using the cofunction rule, state which sine value equals $\cos 75^\circ$.
   (Answer: $\sin 15^\circ$, also $\frac{\sqrt{6} - \sqrt{2}}{4}$.)

5. Find $\tan 75^\circ$ from $\sin 75^\circ$ and $\cos 75^\circ$.
   (Answer: $\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} = 2 + \sqrt{3} \approx 3.7321$.)

6. Verify $\cos 75^\circ$ by the half-angle route from $\cos 150^\circ$.
   (Answer: $\sqrt{\frac{2 - \sqrt{3}}{4}} = \frac{\sqrt{2 - \sqrt{3}}}{2} \approx 0.2588$.)

Where Should You Go Next After Cos 75 Degrees?

Cos 75 degrees is a doorway into the sum-and-difference machinery, and a few natural next steps open from here.

1. Sum and difference identities. The general engine that produced this value, ready to crack $\cos 105^\circ$, $\sin 75^\circ$, and any other split angle.

2. Trigonometric ratios of specific angles. The full first-quadrant family, so the $75^\circ$ row stops feeling like an isolated fact.

3. Cofunction identities. The rule behind $\cos 75^\circ = \sin 15^\circ$, and the fastest way to convert between sine and cosine.

If your child is building these foundations, a live Bhanzu trainer teaches special-angle values starting from the "why" (the unit circle and the sum identities the values are built on) in the Bhanzu trigonometry program.