In right triangle ABC, a = 52 ft and c = 63 ft. Which of the following trig ratios of angle B, would combine the given sides?
cos b
To find the trig ratio that combines the given sides in a right triangle, we need to identify the sides of the triangle in relation to the angle B.
In a right triangle, the side opposite angle B is denoted as the "opposite" side, the side adjacent to angle B is denoted as the "adjacent" side, and the hypotenuse is the longest side, which is always opposite the right angle.
Given that side a = 52 ft and side c = 63 ft, we can identify the sides as follows:
- Side a = 52 ft is the side adjacent to angle B.
- Side c = 63 ft is the hypotenuse of the triangle.
Since we need to find a trigonometric ratio that combines the given sides, we are looking for a ratio that involves the adjacent side (a) and the hypotenuse (c).
The trigonometric ratio that combines the adjacent side and the hypotenuse is the cosine function:
cos(B) = adjacent/hypotenuse
cos(B) = a/c
Substituting the values we have:
cos(B) = 52/63
Hence, the trigonometric ratio of angle B that combines the given sides is cos(B) = 52/63.
To find the trig ratio of angle B, we need to determine the values of sides b and c. We are given that side a is 52 ft and side c is 63 ft.
In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (side c) is equal to the sum of the squares of the other two sides (a and b). Mathematically, it can be written as:
c^2 = a^2 + b^2
Substituting the given values, we have:
63^2 = 52^2 + b^2
Simplifying further:
3969 = 2704 + b^2
To solve for b^2, we subtract 2704 from both sides:
b^2 = 3969 - 2704
b^2 = 1265
To find b, we take the square root of both sides:
b = √1265
Now that we know the values of a and b, we can calculate the trig ratios of angle B.
The sine of angle B is given by the ratio of the length of the side opposite angle B (side a) to the length of the hypotenuse (side c). So, we have:
sin B = a / c
sin B = 52 / 63
The cosine of angle B is given by the ratio of the length of the side adjacent to angle B (side b) to the length of the hypotenuse (side c). So, we have:
cos B = b / c
cos B = √1265 / 63
The tangent of angle B is given by the ratio of the length of the side opposite angle B (side a) to the length of the side adjacent to angle B (side b). So, we have:
tan B = a / b
tan B = 52 / √1265
Therefore, the trig ratios of angle B are:
- sin B = 52 / 63
- cos B = √1265 / 63
- tan B = 52 / √1265