Answer:
- sin30° = 1/2
- cos30° = √3/2
- tan30° = √3/3
Explanation:
To find the values of sin30°, cos30°, and tan30° in a given triangle, we need to use the properties of a special triangle known as a 30-60-90 triangle.
In a 30-60-90 triangle, the angles are in the ratio 1:2:3. The side opposite the 30° angle is the shortest side, the side opposite the 60° angle is the medium side, and the side opposite the 90° angle is the longest side, which is the hypotenuse.
In this case, since we are given the angle of 30°, we can label the sides of the triangle as follows:
- The side opposite the 30° angle is labeled as "x".
- The side opposite the 60° angle is labeled as "x√3" (the length of the shortest side multiplied by the square root of 3).
- The hypotenuse, opposite the 90° angle, is labeled as "2x" (the length of the shortest side multiplied by 2).
Now, let's find the values of sin30°, cos30°, and tan30°:
1. sin30°: In a right triangle, sinθ is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, in our 30-60-90 triangle, sin30° is equal to x divided by 2x, which simplifies to 1/2.
2. cos30°: In a right triangle, cosθ is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In our triangle, the side adjacent to the 30° angle is x√3, and the hypotenuse is 2x. So, cos30° is equal to x√3 divided by 2x, which simplifies to √3/2.
3. tan30°: In a right triangle, tanθ is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In our triangle, the side opposite the 30° angle is x, and the side adjacent to the 30° angle is x√3. So, tan30° is equal to x divided by x√3, which simplifies to 1/√3 or √3/3.
To summarize:
- sin30° = 1/2
- cos30° = √3/2
- tan30° = √3/3
These values can be used to solve various trigonometric problems involving 30° angles in right triangles.