Answer:
The length of the wire is 25 feet. The angle measure formed by the pole and the wire is 36.9 degrees.
Further Explanation
We have two concerns in this case. First, we need to find how long the wire is and second, we need to find what is the measure in degree of the angle formed by the wire and the pole.
To better understand the problem, draw an illustration that represents the problem. See the illustration below. Since we already know that the length of the pole is 20 feet and the distance from the base of the pole to the base of the wire is 50 feet, the Pythagorean Theorem can be used to solve the length of the wire because the shape is a right triangle.
The Pythagorean Theorem states that the square of the longest leg of a right triangle is equal to the sum of the squares of its legs.
[tex]c^2 = a^2+b^2[/tex]
; where [tex]c[/tex] is the longest leg of the right triangle
[tex]a[/tex] and [tex]b[/tex] are the other two legs of the right triangle
Different Type of Triangle
1. Right Triangle
2. Obtuse Triangle
3. Acute Triangle
4. Scalene Triangle
Steps in Solving the Length of the Wire
1. Let [tex]c[/tex] be the length of the wire attached to the top of the pole and on the ground.
2. To solve the length of wire, substitute the value of [tex]a=15[/tex] and [tex]b=20[/tex] into the Pythagorean Theorem.
2. Simplify the right side.
3. Take the square root of each side of the equation.
4. Take the positive value only as length can't be less than zero.
[tex]\begin{aligned}c^2&=15^2+20^\\c^2&=225+400\\c^2&=625\\\sqrt{c^2}&\pm\sqrt{625}\\c&=25\end{aligned}[/tex]
Thus, the length of the wire is 25 feet.
Next is the angle measure made by the wire and the pole. We can solve the angle measure by using one of the trigonometric functions.
Six Trigonometric Functions
1. [tex]cos\:\theta=\frac{\text{adjacent side}}{\text{hypotenuse}}[/tex]
2. [tex]sin\:\theta=\frac{\text{opposite side}}{\text{hypotenuse}}[/tex]
3. [tex]tan\:\theta=\frac{\text{opposite side}}{\text{adjacent side}}[/tex]
4. [tex]sec\:\theta=\frac{\text{hypotensue}}{\text{adjacent side}}[/tex]
5. [tex]csc\:\theta=\frac{\text{hypotensue}}{\text{opposite side}}[/tex]
6. [tex]cot\:\theta=\frac{\text{adjacent side}}{\text{opposite side}}[/tex]
Since we can solve the length of the longest side of the triangle or the hypotenuse, this makes the problem a lot easier because we can use any of the six trigonometric functions to solve for the angle formed by the wire and the pole.
Let's use the first trigonometric function listed above.
Steps in Solving the Angle Measure
1. Substitute the value of the [tex]\text{adjacent side}=20[/tex] and [tex]\text{hypotenuse}=25[/tex] into the cosine function.
2. Solve for the angle measure applying the inverse of the cosine function.
3. Simplify the right side using a scientific calculator.
[tex]\begin{aligned}cos\:\theta&=\frac{\text{adjacent side}}{\text{hypotenuse}}\\cos\:\theta&=\frac{20}{25}\\cos\:\theta&=\frac{4}{5}\\\theta&=cos^{-1}\left(\frac{4}{5}\right)\\\theta&=36.9^o\end{aligned}[/tex]
Thus, the angle measure formed by the wire and the pole is 36.9 degrees
Now assuming that we haven't solved the length of the wire yet. We would still be able to solve the angle measure formed by the wire and the pole. We can use the given details of the problem. Notice that the side opposite to the angle measures 15 feet and the side adjacent to the angle measures 20 feet. Then, we can use the tangent function in this case.
Follow similar steps above in solving the angle measure, however, this time use the tangent function.
[tex]\begin{aligned}tan\:\theta&=\frac{\text{opposite side}}{\text{adjacent side}}\\tan\:\theta&=\frac{15}{20}\\tan\:\theta&=\frac{3}{4}\\\theta&=tan^{-1}\left(\frac{3}{4}\right)\\\theta&=36.9^o\end{aligned}[/tex]
This means that it yields the same angle measure which is 36.9 degrees.
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To learn more about the trigonometric functions, go to https://brainly.ph/question/740865
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