Sketch a diagram describing the given situation. Then recall the definition of the inverse tangent.
Name
Angle of elevation or depression
Kelsey
≈ 4.71^(∘)
Jose
≈ 2.02^(∘)
Practice makes perfect
We are given that Kelsey and Jose are throwing darts from a distance of 8.5 ft and that the center of the bull's-eye on the dartboard is 5.7 ft from the floor. We also know that Jose throws from a height of 6 ft and Kelsey throws from a height of 5 ft. Let's sketch a diagram describing this situation.
We are asked to evaluate the angles of elevation or depression from which each of them must throw to get a bull's-eye. Let's start with Kelsey. Remember that the angle of elevation and the angle of depression are congruent.
Kelsey
Since Kelsey throws from a height of 5 feet, the bull's-eye is 0.7 feet above this height. Let x represents the angle of depression.
To evaluate the value of x, we can use one of the inverse trigonometric ratios. Let's start with recalling that the tangent of ∠ A is the ratio of the leg opposite ∠ A to the leg adjacent ∠ A. Using this definition, we can create an equation for tan x.
tan x=0.7/8.5
Next we will recall that if ∠ A is an acute angle and the tangent of A is a, then the inverse tangent of a is the measure of ∠ A.
Let's rewrite our equation.
tan x=0.7/8.5 ⇓
x=tan^(-1)0.7/8.5
Finally, we will solve the equation using a calculator.
The angle of depression is approximately 4.71^(∘). This means that Kelsey must throw at an angle of approximately 4.71^(∘).
Jose
We know that Jose throws from a height of 6 feet which means that the center of the bull's-eye is 0.3 feet below this height. Let y represent the angle of depression and elevation, as these two angles are always congruent.
To evaluate the value of y, we can use one of the inverse trigonometric ratios. Let's start with recalling that the tangent of ∠ A is the ratio of the leg opposite ∠ A to the leg adjacent ∠ A. Using this definition, we can create an equation for tan y.
tan y=0.3/8.5
Next we will recall that if ∠ A is an acute angle and the tangent of A is a, then the inverse tangent of a is the measure of ∠ A.
Let's rewrite our equation.
tan y=0.3/8.5 ⇓
y=tan^(-1)0.3/8.5
Finally, we will solve the equation using a calculator.
