Cosine, often shortened to cos, is a fundamental trigonometric function that relates an angle of a right triangle to the ratio of its adjacent side and hypotenuse. This article provides a comprehensive overview of cosine, covering its definition, formula, calculation, applications, and relationship to other trigonometric functions.
Understanding the Right Triangle
Before diving into the details of cosine, it’s essential to understand the basics of a right-angled triangle. A right triangle has one angle of 90 degrees. The sides of a right triangle are named as follows:
- Hypotenuse: The longest side, opposite the right angle.
- Opposite: The side opposite to the angle of interest (θ).
- Adjacent: The side adjacent to (next to) the angle of interest (θ), that is not the hypotenuse.
Illustration showing the opposite, adjacent, and hypotenuse sides of a right triangle in relation to an angle.
The position of the “Opposite” and “Adjacent” sides depends on which angle θ you are referencing. The “Hypotenuse” remains the longest side.
Defining Cosine (cos)
Cosine (cos) is one of the primary trigonometric functions, alongside sine (sin) and tangent (tan). In a right triangle, the cosine of an angle (θ) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse:
cos(θ) = Adjacent / Hypotenuse
This means that for any given angle θ, the cosine value represents how the length of the side next to the angle compares to the length of the longest side. This ratio remains constant for a given angle, regardless of the triangle’s size.
Calculating Cosine
To calculate the cosine of an angle, you need to know the lengths of the adjacent side and the hypotenuse. Divide the length of the adjacent side by the length of the hypotenuse.
Example: Finding cos(35°)
A right triangle showing the lengths of the opposite, adjacent, and hypotenuse sides to calculate trigonometric functions for a 35-degree angle.
In this example, we have a right triangle with a 35° angle. The adjacent side has a length of 4.0, and the hypotenuse has a length of 4.9.
cos(35°) = Adjacent / Hypotenuse = 4.0 / 4.9 ≈ 0.82
Therefore, the cosine of 35° is approximately 0.82.
The Impact of Size
An important property of trigonometric ratios like cosine is that the actual size of the triangle doesn’t matter. As long as the angle remains the same, the ratio between the adjacent and hypotenuse will also stay the same. Only a change in the angle will affect the cosine value.
Visual representation of sine, cosine, and tangent as ratios of sides in a right triangle.
Remembering Cosine: Sohcahtoa
A useful mnemonic to remember the relationships between sine, cosine, tangent, and the sides of a right triangle is Sohcahtoa:
- Soh: Sine = Opposite / Hypotenuse
- Cah: Cosine = Adjacent / Hypotenuse
- Toa: Tangent = Opposite / Adjacent
Cosine Values in the Unit Circle
The unit circle, a circle with a radius of 1, is a helpful tool for understanding trigonometric functions for all angles, including those greater than 90 degrees. In the unit circle, the cosine of an angle is represented by the x-coordinate of the point where the terminal side of the angle intersects the circle.
Illustration of a 30-degree right triangle highlighting the side lengths.
Examples of Cosine for Common Angles
Let’s look at some common angles and their cosine values:
Cosine of 30°
In a 30-60-90 triangle, the sides are in the ratio 1:√3:2, where 2 is the length of the hypotenuse, √3 is the length of the adjacent side to the 30° angle, and 1 is the length of the opposite side to the 30° angle. Therefore:
cos(30°) = Adjacent / Hypotenuse = √3 / 2 ≈ 0.866
Cosine of 45°
In a 45-45-90 triangle, the two legs are equal in length, and the sides are in the ratio 1:1:√2, where √2 is the length of the hypotenuse and 1 is the length of each leg. Therefore:
cos(45°) = Adjacent / Hypotenuse = 1 / √2 ≈ 0.707
Applications of Cosine
Cosine, along with sine and tangent, is used to solve real-world problems involving angles and lengths. These functions allow us to:
- Determine unknown angles when we know the lengths of the sides of a right triangle.
- Determine unknown lengths when we know an angle and the length of one side.
Example: Finding Distance with Cosine
Imagine a cable attached to the seabed forms a 39° angle and extends to a length of 30 meters. We can use the cosine function to calculate the horizontal distance from the point where the cable enters the water to the point directly above where it anchors on the seabed.
Since cos(θ) = Adjacent / Hypotenuse, we can rearrange the formula:
Adjacent = cos(θ) * Hypotenuse
Adjacent = cos(39°) * 30 meters
Adjacent ≈ 0.777 * 30 meters ≈ 23.31 meters
Therefore, the horizontal distance is approximately 23.31 meters.
Other Trigonometric Functions
Besides sine, cosine, and tangent, there are three less common trigonometric functions:
- Secant (sec): sec(θ) = Hypotenuse / Adjacent = 1 / cos(θ)
- Cosecant (csc): csc(θ) = Hypotenuse / Opposite = 1 / sin(θ)
- Cotangent (cot): cot(θ) = Adjacent / Opposite = 1 / tan(θ)
