A general equation for the height y of the mountain in terms of d is
y = ( d tan Ф tan Θ ) / ( tan Ф - tan Θ )
Further explanation
First, let us recall the following formula about trigonometry.
Suppose the ΔABC is a right triangle at A, then:
sin ∠A = opposite / hypotenuse
cos ∠A = adjacent / hypotenuse
tan ∠A = opposite / adjacent
Let us now tackle the problem!
The problem can be seen as shown in the picture in the attachment.
Look at ΔABC :
\(\tan \Theta = \frac{opposite}{adjacent}\)
\(\tan \Theta = \frac{AC}{AB}\)
\(\tan \Theta = \frac{y}{x}\)
\(x = \frac{y}{\tan \Theta}\) → Equation 1
Look at ΔADC :
\(\tan \Phi = \frac{opposite}{adjacent}\)
\(\tan \Phi = \frac{AC}{AD}\)
\(\tan \Phi = \frac{y}{x - d}\)
\(y = (x - d)\tan \Phi\)
\(y = (\frac{y}{\tan \Theta} - d)\tan \Phi\) ← Equation 1
\(y = \frac{y ~ \tan \Phi}{\tan \Theta} - d ~ \tan \Phi\)
\(\frac{y ~ \tan \Phi}{\tan \Theta} - y = d ~ \tan \Phi\)
\(y ( \frac{\tan \Phi}{\tan \Theta} - 1) = d ~ \tan \Phi\)
\(y ( \frac{\tan \Phi - \tan \Theta}{\tan \Theta} ) = d ~ \tan \Phi\)
\(\large {\boxed {y = \frac{d ~ \tan \Phi ~ \tan \Theta}{\tan \Phi - \tan \Theta}} }\)
Learn more
- Calculate Angle in Triangle : https://brainly.com/question/12438587
- Periodic Functions and Trigonometry : https://brainly.com/question/9718382
- Trigonometry Formula : https://brainly.com/question/12668178
Answer details
Grade: High School
Subject: Mathematics
Chapter: Trigonometry
Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse
