(KS3, Year 8)
The LessonThe cosine function relates a given angle to the adjacent side and hypotenuse of a right triangle. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
Dictionary DefinitionThe Merriam-Webster dictionary defines the cosine function as "a trigonometric function that for an acute angle is the ratio between the leg adjacent to the angle when it is considered part of a right triangle and the hypotenuse."
In this formula, cos denotes the cosine function, θ is an angle of a right triangle, the adjacent is the length of the side adjacent the angle and the hypotenuse is the length of longest side. The image below shows what we mean:
Interactive WidgetUse this interactive widget to create a right-angled triangle and then use the cosine function to calculate the hidden element. Start by selecting which element you want to hide (using the green buttons) and then clicking in the shaded area.
A Real Example of the Cosine FunctionIt is easier to understand the cosine function with an example.
QuestionFind cos 60° using the right triangle shown below.
Start with the formula:
cos θ = adjacent / hypotenuseDon't forget: / means ÷
Substitute the angle θ, the length of the adjacent and the length of the hypotenuse into the formula. In our example, θ = 60°, the adjacent is 3 cm and the hypotenuse is 6 cm.
cos (60°) = 3 / 6
cos (60°) = 3 ÷ 6
cos (60°) = 0.5
Answer:cos 60° = 0.5.
The Graph of the Cosine FunctionThe cosine function can be plotted on a graph.
Find the angle along the horizontal axis, then go up until you reach the cosine graph. Go across and read the value of cos θ from the vertical axis. We can see from the graph above that cos 60° = 0.5.
The Cosine Function and the Unit CircleThe cosine function can be related to a unit circle, which is a circle with a radius of 1 that is centered at the origin in the Cartesian coordinate system.
For a point at any angle θ, cos θ is given by the x-coordinate of the point.
Lesson SlidesThe slider below gives more information about the cosine function. Open the slider in a new tab
Interactive WidgetHere is an interactive widget to help you learn about the cosine function on a right triangle.
Trigonometry and Right AnglesThe cosine function is a function in trigonometry (called a trigonometric function). The word trigonometry comes from the Greek words 'trigonon' ("triangle") and 'metron' ("measure"). Trigonometry is the branch of mathematics that studies the relationships between the sides and the angles of right triangles. When an angle is defined in a right triangle, the three sides can be defined.
- The side next to the angle is called the adjacent.
- The side opposite the angle is called the opposite.
- The longest side is called the hypotenuse.
Other Trigonometric FunctionsThe cosine function is only one of the trigonometric functions:
The sine function is the ratio of the opposite to the hypotenuse.
The tangent function is the ratio of the opposite to the adjacent.
The Cosine Function and the Unit CircleThe cosine function can be related to the unit circle. Consider a point on the unit circle (x, y). It can be joined to the center by a radius of length 1. A right triangle can be formed under the radius.
The hypotenuse is the radius of 1, the adjacent side has length x and the opposite side length y.
adjacent = cos θ × hypotenuse x = cos θ × 1 x = cos θ
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