Final answer:
In a right triangle with sides of length 6 and 10, the hypotenuse is approximately 11.66. The trigonometric functions are sin a ≈ 0.857, cos a ≈ 0.515, and tan a ≈ 1.667.
Step-by-step explanation:
To find each trigonometric function for angle a in a right triangle with a side adjacent to angle a of length 6 and a side opposite angle a of length 10, we first need to determine the hypotenuse using the Pythagorean theorem. The hypotenuse (h) can be found by the formula: h = √(adjacent side)² + (opposite side)². In this case, h = √6² + 10² = √36 + 100 = √136, which simplifies to h ≈ 11.66.
Once the hypotenuse is known, we can calculate the trigonometric ratios:
- The sine of angle a (sin a) is the ratio of the side opposite angle a to the hypotenuse, so sin a = opposite/hypotenuse = 10/11.66 ≈ 0.857.
- The cosine of angle a (cos a) is the ratio of the side adjacent to angle a to the hypotenuse, so cos a = adjacent/hypotenuse = 6/11.66 ≈ 0.515.
- The tangent of angle a (tan a) is the ratio of the side opposite to angle a to the side adjacent to angle a, so tan a = opposite/adjacent = 10/6 ≈ 1.667.
Therefore, the trigonometric functions of angle a are approximately:
- sin a ≈ 0.857
- cos a ≈ 0.515
- tan a ≈ 1.667