Final answer:
To find the missing side of the right triangle, use the Pythagorean theorem. The exact length of the missing side is √148. To find the exact values of the six trigonometric functions for the angle opposite the shortest side, divide the lengths of the sides of the triangle by the hypotenuse.
Step-by-step explanation:
To find the missing side of the right triangle using the Pythagorean theorem, we can use the formula a² + b² = c², where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. In this case, the given lengths of the legs are 2 cm and 12 cm.
Using the formula, we can substitute the given values of a and b into the equation:
2² + 12² = c²
4 + 144 = c²
148 = c²
To find the exact length of the missing side, we need to take the square root of both sides:
c = √148
Therefore, the exact length of the missing side of the right triangle is √148.
To find the exact values of the six trigonometric functions for the angle opposite the shortest side, we need to divide the lengths of the sides of the triangle by the hypotenuse.
Let's call the angle opposite the shortest side θ.
The six trigonometric functions are:
- sine (sin θ) = opposite/hypotenuse
- cosine (cos θ) = adjacent/hypotenuse
- tangent (tan θ) = opposite/adjacent
- cosecant (csc θ) = 1/sin θ
- secant (sec θ) = 1/cos θ
- cotangent (cot θ) = 1/tan θ
Since the angle opposite the shortest side is θ, we can substitute the values of the sides and hypotenuse into the formulas for each trigonometric function to get the exact values.