Solve for: Hypotenuse Opposite Φ Show all work.
The Correct Answer and Explanation is :
To solve for the hypotenuse when given the opposite side and an angle (Φ), we can apply trigonometric relationships. Specifically, the sine function is used for this type of problem. Here’s the step-by-step explanation:
Step 1: Understanding the Sine Function
In a right triangle, the sine of an angle (Φ) is defined as the ratio of the length of the opposite side to the length of the hypotenuse. This is expressed by the following formula: sin(Φ)=OppositeHypotenuse\sin(\Phi) = \frac{\text{Opposite}}{\text{Hypotenuse}}
Where:
- sin(Φ)\sin(\Phi) is the sine of the angle Φ,
- Opposite is the length of the side opposite to the angle Φ,
- Hypotenuse is the length of the hypotenuse.
Step 2: Rearranging the Formula
To solve for the hypotenuse, we can rearrange the sine formula. Multiply both sides of the equation by the hypotenuse: Hypotenuse=Oppositesin(Φ)\text{Hypotenuse} = \frac{\text{Opposite}}{\sin(\Phi)}
Step 3: Substituting Known Values
At this stage, you would need to substitute the known values for the opposite side and the angle Φ. Let’s assume, for example:
- The opposite side is 5 units.
- The angle Φ=30∘\Phi = 30^\circ.
Thus, we can substitute into the formula: Hypotenuse=5sin(30∘)\text{Hypotenuse} = \frac{5}{\sin(30^\circ)}
Step 4: Calculating the Hypotenuse
We know that sin(30∘)=0.5\sin(30^\circ) = 0.5, so: Hypotenuse=50.5=10\text{Hypotenuse} = \frac{5}{0.5} = 10
Final Answer:
The length of the hypotenuse is 10 units.
Explanation:
This problem involves basic trigonometric principles, specifically the sine function. By applying the formula Hypotenuse=Oppositesin(Φ)\text{Hypotenuse} = \frac{\text{Opposite}}{\sin(\Phi)}, we can find the hypotenuse of a right triangle when we are given the opposite side and an angle. This approach is widely used in various fields such as physics, engineering, and geometry.