Find the value of x using trigonometric ratios. round to the nearest tenth
24-32degrees right triangle
show work!
To solve for x using trigonometric ratios, we need to use the tangent function:
tan(32) = x/24
Multiplying both sides by 24, we get:
x = 24 * tan(32)
Using a calculator, we get:
x ≈ 14.8
Therefore, the value of x is approximately 14.8 units (rounded to the nearest tenth) in the given right triangle.
To find the value of x using trigonometric ratios in a right triangle, we can use the trigonometric functions sine, cosine, or tangent.
Given that we have a right triangle with one angle measuring 32 degrees and an adjacent side measuring 24, we can use the cosine function.
The cosine function is defined as the ratio of the adjacent side to the hypotenuse:
cos(theta) = Adjacent / Hypotenuse
In this case, we are given the adjacent side as 24, and we need to find the hypotenuse, which we'll call x:
cos(32°) = 24 / x
To solve for x, we can rearrange the equation to isolate x:
x = 24 / cos(32°)
Now we can calculate the value of x.
Using a calculator, evaluate the cosine of 32 degrees: cos(32°) ≈ 0.848
Substitute this value back into the equation:
x = 24 / 0.848 ≈ 28.32
Therefore, the value of x, rounded to the nearest tenth, is approximately 28.3.
To find the value of x using trigonometric ratios, we need to use the given information of a right triangle.
The given angle is 32 degrees. Let's label the sides of the triangle as follows:
Opposite side: x
Adjacent side: 24
We are asked to find the value of x, which is the side opposite to the given angle.
Using the trigonometric ratio for the sine function (sin), we can set up the following equation:
sin(32°) = Opposite / Hypotenuse
Since the hypotenuse is not given, we need to find it using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse length is equal to the sum of the squares of the other two sides.
By applying the theorem, we get:
Hypotenuse^2 = x^2 + 24^2
Hypotenuse^2 = x^2 + 576
Now, back to the equation:
sin(32°) = Opposite / Hypotenuse
We can substitute the value of Hypotenuse^2 from the Pythagorean theorem:
sin(32°) = x / sqrt(x^2 + 576)
To solve for x, we can cross-multiply:
x = sin(32°) * sqrt(x^2 + 576)
To simplify this equation, we need to use a calculator to find the value of sin(32°):
sin(32°) ≈ 0.52991927 (rounded to eight decimal places)
Substituting this value back into the equation:
x = 0.52991927 * sqrt(x^2 + 576)
To solve for x, we will use an iterative method (approximation) since this equation is not easily solvable algebraically. Iterative methods involve repeating calculations until convergence is achieved.
Using this iterative method, we can start with an initial guess for x. Let's begin with x = 10:
x = 0.52991927 * sqrt(10^2 + 576)
x ≈ 8.503209 (rounded to six decimal places)
We can repeat the above calculation, using the newly calculated value of x as a replacement for x in the equation, until the value stabilizes or reaches the desired precision.
Continuing this process, after a few iterations, we can approximate the value of x to the nearest tenth:
x ≈ 8.5
Therefore, the value of x, rounded to the nearest tenth, is approximately 8.5.