Cosine is the trigonometric ratio that compares the side adjacent to an angle with the hypotenuse of a right triangle: cos θ = adjacenthypotenuse. On the unit circle, cos θ is the x-coordinate of the point at angle θ. The cosine of an angle equals the sine of its complement.
Cosine is a horizontal distance: the x-coordinate of the point on the unit circle. As the angle opens from 0° to 90°, the point moves left and the cosine shrinks from 1 to 0.
Formal definition and notation
For an acute angleθ of a right triangle, cos θ is the length of the leg adjacent to θ divided by the length of the hypotenuse. For any angle θ in standard position, the terminal side meets the unit circle at (x, y), and cos θ = x. As a function of a real number in radians, y = cosx has range −1 ≤ y ≤ 1, period2π, and is even: cos (−x) = cosx. Sine and cosine are tied together by sin2θ + cos2θ = 1 and by cos θ = sin (90° − θ).
Symbol
Read it as
Meaning
cos θ
cosine theta, or cosine of theta
the cosine of the angle θ
cos−1x
inverse cosine of x, or arccosine of x
the angle from 0° to 180° whose cosine is x; cos−1(0.5) = 60°
Values to know: cos 0° = 1, cos 30° = √32 ≈ 0.866, cos 45° = √22 ≈ 0.707, cos 60° = 12, cos 90° = 0.
From 0° to 90° cosine goes down from 1 to 0, while sine goes up from 0 to 1.
Cofunction rule: cos θ = sin (90° − θ). In a right triangle with acute angles A and B, cosA = sinB.
Cosine is positive in Quadrants I and IV, where x > 0, and negative in Quadrants II and III: cos 120° = −12.
The Pythagorean identity gives one from the other: if sin θ = 35 for an acute angle, then cos θ = √1 − 925 = 45.
The graph y = cosx is the sine wave shifted 90° to the left: it starts at its peak, 1, when x = 0.
Cosine ratio
Cosine ratio
cosθ = adjacenthypotenuse
θ
an acute angle of a right triangle
adjacent
the leg that touches θ
hypotenuse
the longest side, across from the right angle
Rearranged: adjacent = hypotenuse × cos θ, and θ = cos−1(adjacenthypotenuse). In any triangle, the law of cosines c2 = a2 + b2 − 2abcosC uses cosine to connect three sides and one angle.
Cosine examples, step by step
Example 1: Read the cosine from the sides
Warm-up
In a right triangle, the leg next to angle θ is 12, the leg across from it is 5 and the hypotenuse is 13. Find cos θ.
Cosine uses the adjacent side and the hypotenuse: cos θ = adjacenthypotenuse.
The adjacent side is 12, so cos θ = 1213 ≈ 0.92.
Answer:cos θ = 1213.
Example 2: Find a horizontal distance
Core
A 16-foot ramp makes a 25° angle with the ground. How far does it reach horizontally, to the nearest tenth of a foot?
The ramp is the hypotenuse. The horizontal distance touches the 25° angle, so it is the adjacent side.
Adjacent and hypotenuse means cosine: cos 25° = d16.
d = 16 × cos 25° ≈ 16 × 0.9063 ≈ 14.5.
Answer: The ramp reaches about 14.5 feet horizontally.
Example 3: Find an angle with inverse cosine
Test-style
In a right triangle, the side adjacent to angle A is 7 and the hypotenuse is 10. Find angle A to the nearest tenth of a degree.
cosA = 710 = 0.7.
Undo cosine with inverse cosine: A = cos−1(0.7).
A calculator in degree mode gives A ≈ 45.6°.
Answer: Angle A is about 45.6 degrees.
Common mistakes
✗ MistakeUsing the opposite side: cos θ = 513 when the adjacent side is 12.
✓ Insteadcos θ = adjacenthypotenuse = 1213.
CAH: cosine pairs the side that touches the angle with the hypotenuse. The opposite side belongs to sine.
✗ MistakeComputing cos−1(0.5) as 10.5 = 2.
✓ Insteadcos−1(0.5) = 60°, the angle whose cosine is 0.5.
The −1 means inverse function. The reciprocal of cosine is secant, 1cos θ.
✗ MistakeAssuming cos 60° = 2 × cos 30° ≈ 1.73.
✓ Insteadcos 60° = 0.5, which is smaller than cos 30°.
Cosine is not proportional to the angle. On 0 to 90 degrees it gets smaller as the angle grows, and it can never exceed 1.
Sine uses the opposite side (the y-coordinate); cosine uses the adjacent side (the x-coordinate). They swap for the two acute angles of a right triangle: cos 60° = sin 30° = 0.5.
Cosinevs.Secant
Secant is the reciprocal of cosine, sec θ = hypotenuseadjacent. Since cos 60° = 0.5, sec 60° = 2.
Cosinevs.Inverse cosine
Cosine turns an angle into a ratio; inverse cosine, cos−1, turns a ratio back into an angle: cos−1(0) = 90°.
Quick check: 3 questions
Question 1 of 3
What is cos 60°?
Show answer and explanation
Answer: A. On the unit circle the point at 60° is (12, √32), and cosine is the x-coordinate, 12. √32 is sin 60° (or cos 30°), √22 is cos 45°, and 0 is cos 90°.
Question 2 of 3
sin 40° ≈ 0.643. Which value is also about 0.643?
Show answer and explanation
Answer: C.40° and 50° are complementary, so cos 50° = sin 40°. cos 40° and sin 50° both equal about 0.766, and tan 40° ≈ 0.839.
Question 3 of 3
What is cos 120°?
Show answer and explanation
Answer: B.120° is in Quadrant II, where x-coordinates are negative. Its reference angle is 60°, and cos 60° = 12, so cos 120° = −12. 12 forgets the sign, and the √3 choices are sine values.
Cosine: frequently asked questions
What is the difference between sine and cosine?
Both compare a side of a right triangle with the hypotenuse. Sine uses the side opposite the angle and cosine uses the side next to it. On the unit circle, sine is the y-coordinate (height) and cosine is the x-coordinate (horizontal distance). Their graphs are the same wave, with cosine shifted 90° to the left.
What is cos 0?
cos 0° = 1. At an angle of 0, the point on the unit circle is (1, 0), and its x-coordinate is 1. This is the largest value cosine can take. In the other direction, cos 90° = 0, because the point (0, 1) sits straight up with no horizontal distance.
When should you use cosine instead of sine?
Use cosine when the side you know or want touches the angle (the adjacent side) and the other side involved is the hypotenuse. Use sine when the side is across from the angle. For example, to find how far a leaning ladder’s foot is from a wall, given the ladder’s length and its angle with the ground, use cosine.
Acute AngleAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5, Grade 7Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Unit CircleThe unit circle is the circle with radius 1 centered at the origin, x2 + y2 = 1. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ). This lets sine and cosine work for every angle, including obtuse and negative angles, not just the acute angles of right triangles.Trigonometry · Algebra 2, PrecalculusFunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1RadianA radian is a unit for measuring angles: one radian is the central angle that cuts off an arc exactly as long as the circle’s radius. A full turn is 2π radians, so π radians = 180° and 1 radian is about 57.3°. To change degrees to radians, multiply by π180.Trigonometry · Geometry, Algebra 2, PrecalculusPeriodThe period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sinx and y = cosx have period 2π, or 360°, and y = tanx has period π. For y = sin(Bx) or y = cos(Bx), the period is 2π|B|.Trigonometry · Algebra 2, PrecalculusSineSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2, PrecalculusReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7QuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7FootA foot is a US customary unit of length equal to 12 inches, about the length of a standard school ruler. Its abbreviation is ft, and its plural is feet. There are 3 feet in a yard and 5,280 feet in a mile, and 1 foot is exactly 30.48 centimeters.Measurement · Grade 2, Grade 3, Grade 4AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1TrigonometryTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2, PrecalculusTangentTangent (tan) is the trigonometric ratio of the side opposite an acute angle to the side adjacent to it in a right triangle: tan θ = oppositeadjacent. For legs 3 and 4, tan θ = 34 = 0.75. It is the TOA in SOH CAH TOA, and it equals sin θcos θ.Trigonometry · Geometry, Algebra 2, Precalculus