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Cosine (cos θ): Definition, Formula, Graph & Values

cosine
cos

The trig function cosine, which is written cos θ. For acute angles, cos θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = cos x is a periodic function with period 2π.

Left: right triangle with hypotenuse, adjacent, and opposite sides labelled; cos θ = adjacent/hypotenuse. Right: unit circle showing cos θ = x/r at a point on the circle.

Key Formula

cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}
Where:
  • θ\theta = The angle in the right triangle you are evaluating
  • adjacent\text{adjacent} = The length of the side next to angle θ (not the hypotenuse)
  • hypotenuse\text{hypotenuse} = The length of the longest side of the right triangle, opposite the right angle

Worked Example

Problem: A right triangle has an adjacent side of 4, a hypotenuse of 5, and an angle θ between them. Find cos θ.
Step 1: Identify the sides relative to angle θ. The side next to θ (adjacent) is 4, and the hypotenuse is 5.
adjacent=4,hypotenuse=5\text{adjacent} = 4, \quad \text{hypotenuse} = 5
Step 2: Apply the cosine ratio from SOHCAHTOA.
cos⁡θ=adjacenthypotenuse=45\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}
Step 3: Simplify the fraction to a decimal if needed.
cos⁡θ=0.8\cos\theta = 0.8
Answer: cos θ = 4/5 = 0.8

Another Example

This example uses the unit circle definition for an angle beyond 90°, where SOHCAHTOA alone does not apply. It shows how the sign of cosine depends on the quadrant.

Problem: Find the exact value of cos 120°.
Step 1: Convert 120° to radians or locate it on the unit circle. 120° lies in the second quadrant.
120°=2π3120° = \frac{2\pi}{3}
Step 2: Find the reference angle. The reference angle for 120° is 180° − 120° = 60°.
reference angle=60°\text{reference angle} = 60°
Step 3: Recall that cos 60° = 1/2. Since cosine is negative in the second quadrant, attach a negative sign.
cos⁡60°=12\cos 60° = \frac{1}{2}
Step 4: Write the final result.
cos⁡120°=−12\cos 120° = -\frac{1}{2}
Answer: cos 120° = −1/2

Frequently Asked Questions

What is the difference between cosine and sine?
Sine gives the ratio of the opposite side to the hypotenuse, while cosine gives the ratio of the adjacent side to the hypotenuse. On the unit circle, cos θ is the x-coordinate of the point and sin θ is the y-coordinate. Their graphs have the same shape but are shifted: cos x = sin(x + π/2).
When is cosine negative?
Cosine is negative when the angle's terminal side lies in the second or third quadrant — that is, for angles between 90° and 270° (or π/2 and 3π/2 radians). A quick way to remember this: cosine corresponds to the x-coordinate on the unit circle, and x is negative on the left half of the circle.
What is the cosine of 0, 90, and 180 degrees?
cos 0° = 1, cos 90° = 0, and cos 180° = −1. These are key reference values worth memorizing. On the unit circle, they correspond to the points (1, 0), (0, 1), and (−1, 0), respectively.

Cosine (cos) vs. Sine (sin)

Cosine (cos)Sine (sin)
Right-triangle ratioAdjacent / HypotenuseOpposite / Hypotenuse
Unit circle coordinatex-coordinate of the pointy-coordinate of the point
Value at 0°10
Value at 90°01
Graph shiftcos x = sin(x + π/2)sin x = cos(x − π/2)
Period2π2π

Why It Matters

Cosine appears constantly in trigonometry courses, from solving right triangles to graphing sinusoidal functions and verifying identities. In physics, it is essential for breaking forces into components, modeling wave motion, and computing work done by a force at an angle. Standardized tests like the SAT and ACT regularly include problems that require you to evaluate or apply cosine.

Common Mistakes

Mistake: Confusing the adjacent and opposite sides when computing cos θ.
Correction: Always identify sides relative to the specific angle θ you are working with. The adjacent side is the leg that forms part of angle θ; the opposite side is the leg across from θ. Mixing them up gives you sine instead of cosine.
Mistake: Forgetting to check the quadrant and assigning the wrong sign to the result.
Correction: Cosine is positive in quadrants I and IV, and negative in quadrants II and III. After finding the cosine of the reference angle, determine which quadrant the original angle falls in and apply the correct sign.

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