Right triangle trigonometry is the study of the relationships between the sides and angles of a right triangle. Here are some key concepts in right triangle trigonometry:

- The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides.
- Trigonometric ratios: The sine, cosine, and tangent of an angle in a right triangle can be expressed as ratios of the lengths of the sides. The sine of the angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. The cosine of the angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. The tangent of the angle is equal to the length of the side opposite the angle divided by the length of the side adjacent to the angle.
- Special right triangles: Some right triangles have side lengths that are in a fixed ratio to each other. These triangles are called “special right triangles” and include 30-60-90 triangles and 45-45-90 triangles.
- Applications: Right triangle trigonometry has many practical applications, including finding distances using triangulation, surveying, and navigation. It is also used in engineering, physics, and other fields.

Here are some exercises you can include on a right triangle trigonometry worksheet:

- Find the missing side length in a right triangle using the Pythagorean theorem.
- Find the sine, cosine, and tangent of an angle in a right triangle given the lengths of the sides.
- Solve for an unknown angle in a right triangle using the trigonometric ratios.
- Determine whether a given triangle is a right triangle, and if so, find the lengths of the sides using the Pythagorean theorem.
- Find the area of a right triangle given the lengths of the sides.
- Use the trigonometric ratios to find the missing side length in a 30-60-90 triangle or a 45-45-90 triangle.
- Solve a word problem involving a right triangle, such as finding the distance between two points using triangulation.
- Use the trigonometric ratios to find the slope of a line given the coordinates of two points on the line.

### Sample Test Questions

- In a right triangle, the length of the hypotenuse is 5 and the length of one of the other sides is 3. What is the length of the other side?
- In a right triangle, the length of the hypotenuse is 8 and the length of one of the other sides is 15. What is the sine of the angle opposite the side with length 15?
- In a right triangle, the length of the hypotenuse is 12 and the length of one of the other sides is 16. What is the cosine of the angle adjacent to the side with length 16?
- In a right triangle, the length of one side is 8 and the length of the other side is 15. What is the tangent of the angle opposite the side with length 8?
- A right triangle has an angle with a measure of 30 degrees. What is the ratio of the length of the side opposite the angle to the length of the hypotenuse?
- A right triangle has an angle with a measure of 45 degrees. What is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse?
- A right triangle has an area of 36 and a hypotenuse with length 12. What is the length of the other side?
- In a right triangle, the length of the hypotenuse is 13 and the length of one of the other sides is 5. What is the measure of the angle opposite the side with length 5?
- In a 45-45-90 triangle, the length of one of the legs is 6. What is the length of the hypotenuse?
- In a 30-60-90 triangle, the length of the hypotenuse is 12. What is the length of the shorter leg?

#### Answer

- The length of the other side is 4.
- The sine of the angle opposite the side with length 15 is 0.6.
- The cosine of the angle adjacent to the side with length 16 is 0.8.
- The tangent of the angle opposite the side with length 8 is 1.6.
- The ratio of the length of the side opposite the angle to the length of the hypotenuse is 0.6.
- The ratio of the length of the side adjacent to the angle to the length of the hypotenuse is 0.7071067811865475.
- The length of the other side is 6.
- The measure of the angle opposite the side with length 5 is 53.13010235415598 degrees.
- The length of the hypotenuse is 8.48528137423857.
- The length of the shorter leg is 6.