Showing relationship between cosine and sine of complements | Trigonometry | Khan Academy

Recent questions

• What is a right triangle?

  A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This specific angle is known as the right angle, and it distinguishes right triangles from other types of triangles, such as acute and obtuse triangles, which have angles less than or greater than 90 degrees, respectively. In a right triangle, the sides are categorized into three types: the hypotenuse, which is the longest side opposite the right angle, and the two legs, which are the sides that form the right angle. Right triangles are fundamental in geometry and trigonometry, as they provide a basis for various mathematical principles and theorems, including the Pythagorean theorem, which relates the lengths of the sides of the triangle.

• How do you calculate sine?

  The sine of an angle in a right triangle is calculated as the ratio of the length of the side opposite the angle to the length of the hypotenuse. This relationship can be expressed mathematically as sin(θ) = opposite/hypotenuse, where θ represents the angle in question. For example, if you have a right triangle where the length of the side opposite angle A is 3 units and the length of the hypotenuse is 5 units, the sine of angle A would be calculated as sin(A) = 3/5, which equals 0.6. This ratio is crucial in various applications, including physics, engineering, and computer graphics, as it helps in determining the height of objects, angles of elevation, and more.

• What are complementary angles?

  Complementary angles are two angles whose measures add up to 90 degrees. In the context of a right triangle, the two non-right angles are always complementary, meaning if one angle measures θ degrees, the other angle will measure 90 - θ degrees. This relationship is significant in trigonometry, as it leads to the identity that the sine of an angle is equal to the cosine of its complementary angle. For instance, if angle A is 30 degrees, then angle B, being complementary, would be 60 degrees, and it holds that sin(30°) = cos(60°). Understanding complementary angles is essential for solving various geometric problems and for applying trigonometric functions effectively.

• What is cosine in a triangle?

  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. This can be expressed mathematically as cos(θ) = adjacent/hypotenuse, where θ is the angle in question. For example, if the length of the side adjacent to angle A is 4 units and the hypotenuse is 5 units, then the cosine of angle A would be calculated as cos(A) = 4/5, which equals 0.8. The cosine function is widely used in various fields, including physics, engineering, and computer science, as it helps in determining distances, angles, and other critical measurements in both theoretical and practical applications.

• What is the relationship between sine and cosine?

  The relationship between sine and cosine is defined by the complementary angle identity, which states that the sine of an angle is equal to the cosine of its complement. In mathematical terms, this can be expressed as sin(θ) = cos(90° - θ). For example, if you take an angle of 45 degrees, its complement is also 45 degrees, and it holds that sin(45°) = cos(45°). This relationship is fundamental in trigonometry and is used to simplify calculations and solve problems involving angles. It also highlights the symmetry in the unit circle, where the coordinates of points correspond to sine and cosine values, making it easier to understand the behavior of trigonometric functions across different quadrants.