find the angle θ in the right triangle in the following problem. Round to the nearest degree, if necessary.
cos θ = 0.8192 A = 45 H = ?
is A adjacent to or opposite θ?
Adjacent
To find the angle θ in the right triangle, we can use the inverse cosine function since we know the value of the cosine of θ.
Given that cos θ = 0.8192, we can find θ by taking the inverse cosine of 0.8192.
θ = cos^(-1)(0.8192)
Using a calculator or a trigonometric table, we find that θ is approximately 35 degrees (rounded to the nearest degree).
Now, let's find the length of the hypotenuse (H) in the right triangle.
We are given that A = 45 (one of the acute angles) and we need to find H.
In a right triangle, the sine function relates the lengths of the sides to the acute angle. The sine of an acute angle A is defined as the ratio of the length of the opposite side to the hypotenuse.
Using the sine function:
sin A = Opposite / Hypotenuse
sin 45 = Opposite / H
Since sin 45 = 1/√2, we have:
1/√2 = Opposite / H
Now, we can solve for H:
H = Opposite / (1/√2)
H = Opposite * √2
As the opposite side of a 45-degree angle is equal to the adjacent side, we have:
H = A * √2
H = 45 * √2
Simplifying this expression, we have:
H ≈ 63.64
Therefore, the length of the hypotenuse (H) is approximately 63.64.
To find the angle θ in the right triangle, you can use the inverse cosine function. The inverse cosine function or arc cosine, denoted as cos^(-1)(x) or acos(x), gives you the angle whose cosine value is x.
In this problem, you are given the cosine value, cos θ = 0.8192. To find the angle θ, you can use the inverse cosine function:
θ = cos^(-1)(0.8192)
Now, let's calculate the value of θ using a calculator:
θ ≈ 35.31 degrees
So, the angle θ in the right triangle is approximately 35 degrees (rounded to the nearest degree).