Basic trigonometry II | Basic trigonometry | 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 creates a unique relationship between the lengths of the sides, which is fundamental in trigonometry. The side opposite the right angle is known as the hypotenuse, while the other two sides are referred to as the legs. The properties of right triangles allow for the application of the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship is crucial for calculating distances and angles in various fields, including physics, engineering, and architecture.

• How do you calculate sine?

  Sine is calculated as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. This relationship can be expressed mathematically as \( \text{sine}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). For example, if you have a right triangle where the length of the side opposite the angle \( \theta \) is 7 units and the hypotenuse is \( \sqrt{65} \), the sine of angle \( \theta \) would be \( \frac{7}{\sqrt{65}} \). This ratio is essential in trigonometry and is used to solve various problems involving angles and distances.

• What is the Pythagorean theorem?

  The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the formula \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. This theorem is widely used in various applications, including construction, navigation, and physics, to determine distances and relationships between different geometric figures.

• What is cosine used for?

  Cosine is a trigonometric function that is used to relate the angle of a right triangle to the lengths of its sides. Specifically, cosine is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. This can be expressed mathematically as \( \text{cosine}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). Cosine is particularly useful in various applications, such as calculating angles in engineering, physics, and computer graphics. For instance, if you know the lengths of the sides of a right triangle, you can use cosine to find the angle, which is essential for solving problems involving forces, motion, and design.

• What does "soh cah toa" mean?

  "Soh cah toa" is a mnemonic device used to remember the definitions of the three primary trigonometric functions: sine, cosine, and tangent. Each part of the phrase corresponds to a specific ratio in a right triangle. "Soh" stands for sine, which is the ratio of the opposite side to the hypotenuse (\( \text{sine} = \frac{\text{opposite}}{\text{hypotenuse}} \)). "Cah" represents cosine, the ratio of the adjacent side to the hypotenuse (\( \text{cosine} = \frac{\text{adjacent}}{\text{hypotenuse}} \)). Finally, "toa" refers to tangent, which is the ratio of the opposite side to the adjacent side (\( \text{tangent} = \frac{\text{opposite}}{\text{adjacent}} \)). This mnemonic is a helpful tool for students and professionals alike when learning or applying trigonometric concepts.