For a given right triangle, angle A = 61± and side b = 39 feet. What is the length of side a to the nearest foot?
12
To find the length of side a in the given right triangle, we need to use the trigonometric functions sine, cosine, or tangent. However, we don't have enough information to directly apply any of these functions since we only know the measure of one angle and one side.
To proceed, we can use the definition of sine or cosine to find the value of the missing side. Since we know side b, we can try using sine:
sin(A) = opposite/hypotenuse
In this case, side b is the opposite side of angle A and the hypotenuse is the unknown side a. Rearranging the equation, we get:
a = b/sin(A)
Now, let's plug in the values:
b = 39 feet
A = 61° (or -61°)
To solve for a, we need to consider both the positive and negative values of angle A since we have a ± sign. Let's calculate both cases:
For A = 61°:
a = 39/sin(61°)
Using a scientific calculator or an online calculator that supports trigonometric functions, we can find the value of sin(61°) which is approximately 0.8746. Substituting this value into the equation, we get:
a = 39/0.8746 ≈ 44.60 feet
So, when angle A is 61°, the length of side a is approximately 44.60 feet.
For A = -61°:
a = 39/sin(-61°)
Since sine is an odd function, sin(-61°) = -sin(61°). Plugging in the value of sin(61°) we obtained earlier, we get:
a = 39/-0.8746 ≈ -44.60 feet
Even though we have a negative value, the magnitude is still the same. Therefore, when angle A is -61°, the length of side a is approximately 44.60 feet.
In both cases, we rounded to the nearest foot, so the length of side a is approximately 45 feet.