Cosine Calculator
Calculate cosine instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods. Get results you can export or share.
Formula
cos(x) = adjacent / hypotenuse
The cosine of angle x equals the ratio of the adjacent side to the hypotenuse in a right triangle. On the unit circle, cos(x) is the x-coordinate of the point at angle x.
Worked Examples
Example 1: Cosine of 60 Degrees
Problem: Calculate cos(60 degrees) and verify using the Pythagorean identity.
Solution: cos(60) = 0.5\nsin(60) = 0.86602540 (sqrt(3)/2)\nPythagorean check: sin2(60) + cos2(60) = 0.75 + 0.25 = 1.0 (confirmed)\nsec(60) = 1/cos(60) = 1/0.5 = 2.0
Result: cos(60) = 0.5 | sin(60) = 0.86602540 | Identity verified
Example 2: Cosine of 150 Degrees
Problem: Find cos(150 degrees) and identify the quadrant and reference angle.
Solution: 150 degrees is in Quadrant II (90-180 degrees).\nReference angle = 180 - 150 = 30 degrees.\ncos(150) = -cos(30) = -sqrt(3)/2 = -0.86602540\nCosine is negative in Q2, confirmed.\nsin(150) = sin(30) = 0.5
Result: cos(150) = -0.86602540 | Quadrant II | Reference angle = 30
Frequently Asked Questions
What is the cosine function in trigonometry?
The cosine function is one of the three primary trigonometric functions, along with sine and tangent. In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Written as cos(x) = adjacent/hypotenuse, it measures how much of the hypotenuse lies along the horizontal direction relative to the angle. The cosine function produces values ranging from -1 to 1, oscillating smoothly as the angle changes. It is a fundamental building block of mathematics with applications spanning physics, engineering, computer science, signal processing, and virtually every quantitative field.
What are the cosine values for common angles?
The cosine values for standard angles are worth memorizing. cos(0) = 1, cos(30) = sqrt(3)/2 which is approximately 0.866, cos(45) = sqrt(2)/2 which is approximately 0.707, cos(60) = 1/2 = 0.5, cos(90) = 0, cos(120) = -1/2, cos(180) = -1, cos(270) = 0, and cos(360) = 1. A helpful pattern to remember is that cosine starts at 1, decreases to 0 at 90 degrees, reaches -1 at 180 degrees, returns to 0 at 270 degrees, and completes its cycle back at 1 for 360 degrees. These values form the foundation for solving most trigonometric problems without a calculator.
How is cosine related to the unit circle?
On the unit circle (a circle with radius 1 centered at the origin), the cosine of an angle equals the x-coordinate of the point where the angle's terminal side intersects the circle. If you draw an angle from the positive x-axis, the point on the circle at that angle has coordinates (cos(x), sin(x)). This geometric interpretation extends cosine beyond acute angles to all real numbers: as you sweep around the circle, the x-coordinate oscillates between -1 and 1. Cosine is positive in the first and fourth quadrants (right half of circle) and negative in the second and third quadrants (left half). This unit circle definition is the modern foundation for understanding all trigonometric functions.
What is the difference between cosine and inverse cosine?
Cosine takes an angle as input and produces a ratio as output, while inverse cosine (arccos or cos inverse) does the opposite: it takes a ratio as input and returns the angle. For example, cos(60) = 0.5 means the cosine of 60 degrees is one-half, while arccos(0.5) = 60 means the angle whose cosine is one-half is 60 degrees. The inverse cosine function has a restricted range of 0 to 180 degrees (0 to pi radians) to ensure it returns a single unique value. This is because cosine is not one-to-one over its full domain. The distinction between these functions is crucial for solving equations where you need to find an unknown angle from a known ratio.
How is cosine used in the Pythagorean identity?
The fundamental Pythagorean identity states that sin squared(x) plus cos squared(x) equals 1, written as sin2(x) + cos2(x) = 1. This identity holds true for every possible angle and is derived directly from the Pythagorean theorem applied to a right triangle with hypotenuse 1. From this primary identity, you can derive two secondary forms: dividing everything by cos2(x) gives 1 + tan2(x) = sec2(x), and dividing by sin2(x) gives cot2(x) + 1 = csc2(x). The Pythagorean identity is arguably the most important equation in trigonometry, used constantly in simplifying expressions, proving other identities, solving equations, and performing calculus operations like integration by substitution.
What is the cosine rule (law of cosines)?
The law of cosines generalizes the Pythagorean theorem to work with any triangle, not just right triangles. It states c2 = a2 + b2 - 2ab cos(C), where a, b, and c are side lengths and C is the angle opposite side c. When C = 90 degrees, cos(90) = 0 and the formula reduces to the Pythagorean theorem c2 = a2 + b2. The law of cosines is used to find a missing side when you know two sides and the included angle (SAS), or to find a missing angle when you know all three sides (SSS). It is essential in surveying, navigation, physics force calculations, and any situation involving non-right triangles where you need to relate sides and angles.