Math
Cosine Calculator
Cosine is the ratio adjacent / hypotenuse in a right triangle and the x-coordinate on the unit circle.
This cosine calculator evaluates cos θ in degrees or radians with exact answers such as cos 60° = 1/2 and cos 30° = √3/2, solves cos θ = x for every angle, and finds a missing right-triangle side or angle.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
cosine result
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- Angle in degrees
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- Angle in radians
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- Quadrant
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- Reference angle
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- Reciprocal (sec)
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- Cofunction / ratio
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More details (1 more)
- More
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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What cosine measures
In a right triangle, cosine compares the side next to an acute angle with the The longest side of a right triangle, always opposite the right angle.. On the unit circle it is the horizontal position (x-coordinate) of the point at angle θ, so cosine is positive on the right half of the circle and negative on the left half.
Exact cosine values
| θ | cos θ | Decimal |
|---|---|---|
| 0° | 1 | 1 |
| 30° (π/6) | √3/2 | 0.8660 |
| 45° (π/4) | √2/2 | 0.7071 |
| 60° (π/3) | 1/2 | 0.5 |
| 90° (π/2) | 0 | 0 |
| 120° (2π/3) | -1/2 | -0.5 |
| 150° (5π/6) | -√3/2 | -0.8660 |
| 180° (π) | -1 | -1 |
| 270° (3π/2) | 0 | 0 |
Four ways to use this calculator
| Mode | You enter | You get |
|---|---|---|
| Find cos θ | An angle in any unit | cos θ exact and decimal, sec θ, sin of the complement, quadrant, reference angle |
| Find θ (arccos) | A value between -1 and 1 | Principal angle (0° to 180°), all angles in [0°, 360°), general solution |
| Right-triangle side | An acute angle and the adjacent side or hypotenuse | The other of the two, plus the third side and other angle |
| Right-triangle angle | Adjacent side and hypotenuse | The angle, opposite side, and other angle |
How to read the answer
- cos θ is positive in One of the four regions the x- and y-axes cut the coordinate plane into. I and IV and negative in Quadrants II and III.
- It equals 1 at 0°, 0 at 90° and 270°, and -1 at 180°.
- Cosine is an even function: cos(-45°) = cos 45° = √2/2.
- The cofunction line shows sin(90° − θ), which always equals cos θ.
Common mistakes and edge cases
- cos 60 in An angle unit where a full turn is 2π instead of 360 degrees. mode is -0.952, not 0.5. Check the unit selector.
- cos θ = -1.5 has no solution; the calculator reports the domain error.
- arccos returns angles from 0° to 180° only. For cos θ = 0.5 the second solution is 360° − 60° = 300°.
- The adjacent side touches the angle but is not the hypotenuse.
Worked examples
cos 60°
cosine result
1/2 (0.5)
Quadrant II cosine
cos 150° = -√3/2
cosine result
-√3/2 (-0.866025)
Negative angle
cos(-π/4) = √2/2
cosine result
√2/2 (0.707107)
cos 90°
zero case
cosine result
0
Solve cos θ = -0.5
cosine result
120° = 2π/3
Impossible cosine
cosine result
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Find the adjacent side
θ = 60°, hypotenuse = 10
cosine result
Adjacent side = 5
Find the hypotenuse
θ = 45°, adjacent = 7
cosine result
Hypotenuse = 9.899495
Angle from two sides
adjacent 4, hypotenuse 5
cosine result
θ = 36.8699°
Adjacent longer than hypotenuse
cosine result
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Frequently asked questions
What is cos 60 degrees?+
cos 60° = 1/2 exactly. In a 30-60-90 triangle the side next to the 60° angle is half the hypotenuse.
Why is cos 120° negative?+
120° is in Quadrant II, where the x-coordinate on the unit circle is negative. Its reference angle is 60°, so cos 120° = -cos 60° = -1/2.
Is cos(-θ) equal to cos θ?+
Yes. Cosine is even: reflecting an angle below the x-axis does not change the x-coordinate. cos(-30°) = cos 30° = √3/2.
How do I find the adjacent side with cosine?+
adjacent = hypotenuse × cos θ. If instead you know the adjacent side and need the hypotenuse, divide: hypotenuse = adjacent / cos θ.
What is the inverse of cosine?+
Arccosine (cos⁻¹). It returns the angle between 0° and 180° whose cosine is the given value; the other solution in one turn is 360° minus that angle.
When is cosine zero?+
At 90°, 270°, and every angle that differs from them by a multiple of 180°. Those are exactly the angles where tangent and secant are undefined.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026