To find the values of all six trigonometric functions of the right triangle ABC with sides a = 7, b = 24, and c = 25, we start by identifying the relationships between the angles and the sides.
In a right triangle:
- Sine of an angle (sin) is the ratio of the length of the opposite side to the hypotenuse.
- Cosine of an angle (cos) is the ratio of the length of the adjacent side to the hypotenuse.
- Tangent of an angle (tan) is the ratio of the length of the opposite side to the adjacent side.
Let’s denote:
- Angle A (opposite side a = 7)
- Angle B (opposite side b = 24)
- Angle C (the right angle, which is 90 degrees)
Now, we can calculate each of the trigonometric functions:
For angle A:
- sin(A) = opposite/hypotenuse = a/c = 7/25
- cos(A) = adjacent/hypotenuse = b/c = 24/25
- tan(A) = opposite/adjacent = a/b = 7/24
For angle B:
- sin(B) = opposite/hypotenuse = b/c = 24/25
- cos(B) = adjacent/hypotenuse = a/c = 7/25
- tan(B) = opposite/adjacent = b/a = 24/7
The six trigonometric function values are:
- sin(A) = 7/25
- cos(A) = 24/25
- tan(A) = 7/24
- sin(B) = 24/25
- cos(B) = 7/25
- tan(B) = 24/7
This way, we’ve found all six trigonometric functions for the triangle ABC!