Final answer:
In high school mathematics, we can use the cosine, sine, and tangent ratios to find missing sides of right triangles. By knowing one side length and the angle, we calculate others using trigonometric relations and the Pythagorean theorem.
Step-by-step explanation:
Finding Missing Sides in Right Triangles Using Trigonometric Ratios
To find the lengths of the missing sides in right triangles, we can use trigonometric ratios such as sine, cosine, and tangent. These ratios relate the lengths of the sides of a right triangle to the angles. The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse, while the sine is the ratio of the opposite side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.
Here are examples using the cosine, sine, and tangent ratios to find missing side lengths:
- For cosB = 4/5, a=10, we can calculate the hypotenuse c using the relation cosB = a/c. Rearranging for c, we get c = a / cosB = 10 / (4/5) = 12.5.
- If the sinB = 1/2 and the hypotenuse c is 20, the opposite side a can be found using a = c * sinB = 20 * (1/2) = 10.
- With tanA = 5/12, a = 20, we can solve for the adjacent side b using b = a / tanA = 20 / (5/12) = 48.
- Lastly, for tanA = 1/3, a = 2, the adjacent side b can be calculated as b = a / tanA = 2 / (1/3) = 6.
Note that we also use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a² + b² = c².