Use a table of trigonometric values to 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 = ?
To find the angle θ in the right triangle, we can use the inverse cosine function, also known as arccosine.
Since cos θ = 0.8192, we can use a table of trigonometric values or a calculator to find the angle whose cosine is 0.8192.
From the table, we find that the angle whose cosine is 0.8192 is approximately 36 degrees (to the nearest degree).
Therefore, θ ≈ 36 degrees.
However, the given values of A = 45 and H = ? are not sufficient to find the length of the hypotenuse. We need either the length of the angle or another side length to determine the length of the hypotenuse (H).
To find the angle θ, we can use the inverse cosine function (cos^-1).
First, look for the angle on the table of trigonometric values that has a cosine value of 0.8192. Since cosine is positive, we are looking for an acute angle.
On the table, find the value closest to 0.8192. The closest value is 0.820. The corresponding angle is approximately 36°.
Therefore, θ ≈ 36°.
Now, let's find the length of the hypotenuse (H) in the right triangle.
Using the Pythagorean theorem, we know that the square of the hypotenuse (H) is equal to the sum of the squares of the other two sides.
H^2 = A^2 + B^2
Substituting the given values, A = 45 (one of the legs of the triangle), and θ = 36° (angle opposite to side A):
H^2 = 45^2 + B^2
H^2 = 2025 + B^2
To solve for H, we need to find the value of B. We can use the trigonometric ratio sine (sin).
sin(θ) = B/H
sin(36°) = B/H
We need to rearrange the equation to solve for B:
B = sin(36°) * H
Using a calculator or trigonometric table, we find that sin(36°) ≈ 0.5878.
Substituting this value into the equation:
B ≈ 0.5878 * H
Now, we can substitute this expression for B into the previous equation:
H^2 = 2025 + (0.5878 * H)^2
Simplifying:
H^2 = 2025 + 0.3455 * H^2
H^2 - 0.3455 * H^2 = 2025
0.6545 * H^2 = 2025
Dividing both sides by 0.6545:
H^2 ≈ 3097.12
Taking the square root of both sides:
H ≈ √3097.12
H ≈ 55.68
Therefore, the length of the hypotenuse, H, is approximately 55.68.