a We are asked to determine the angle of elevation to Mount Whitney if the horizontal distance from the base to the peak is 1200 meters and we know that Mount Whitney is 2530 meters above the ground. Let's take a look at the given picture.
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=2530/1200
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=2530/1200 ⇓
x=tan^(-1)2530/1200
Finally, we will solve the equation using a calculator.
The angle of elevation to Mount Whitney is approximately 64.6^(∘).
b In this part we are asked to evaluate the horizontal distance from sea level to Death Valley if the angle of depression is 38^(∘) and Death Valley is 86 meters below sea level. Let's take a look at the given picture.
To find the value of x, we can use one of the trigonometric ratios. Let's recall 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 38^(∘).
tan 38^(∘)=86/x
Let's solve the above equation.
