If x = psinA cosB, y = psinA sinB and z = pcos A, then what is the value of x 2 + y 2 + z 2 ?
p 2
Understanding the Problem: Finding the Value of x2 + y2 + z2
The question asks us to find the value of the expression \(x^2 + y^2 + z^2\) given the values of x, y, and z in terms of parameters p, A, and B. We are given:
- \(x = p \sin A \cos B\)
- \(y = p \sin A \sin B\)
- \(z = p \cos A\)
To solve this, we need to calculate the square of each term (x, y, and z) and then add them together. This will involve using some fundamental trigonometric identities.
Step-by-Step Calculation of x2 + y2 + z2
First, let's calculate the square of each given expression:
- Calculate \(x^2\):
\(x^2 = (p \sin A \cos B)^2\) \(x^2 = p^2 \sin^2 A \cos^2 B\) - Calculate \(y^2\):
\(y^2 = (p \sin A \sin B)^2\) \(y^2 = p^2 \sin^2 A \sin^2 B\) - Calculate \(z^2\):
\(z^2 = (p \cos A)^2\) \(z^2 = p^2 \cos^2 A\)
Now, we need to find the sum \(x^2 + y^2 + z^2\):
\(x^2 + y^2 + z^2 = (p^2 \sin^2 A \cos^2 B) + (p^2 \sin^2 A \sin^2 B) + (p^2 \cos^2 A)\)
We can factor out the common term \(p^2\) from all terms:
\(x^2 + y^2 + z^2 = p^2 (\sin^2 A \cos^2 B + \sin^2 A \sin^2 B + \cos^2 A)\)
Now, let's look at the terms inside the parenthesis. The first two terms, \(\sin^2 A \cos^2 B\) and \(\sin^2 A \sin^2 B\), have \(\sin^2 A\) in common. We can factor that out:
\(x^2 + y^2 + z^2 = p^2 [\sin^2 A (\cos^2 B + \sin^2 B) + \cos^2 A]\)
Here, we can use the fundamental trigonometric identity: \(\cos^2 \theta + \sin^2 \theta = 1\). Applying this to the terms inside the inner parenthesis (\(\cos^2 B + \sin^2 B\)), we get:
\(\cos^2 B + \sin^2 B = 1\)
Substitute this back into our expression:
\(x^2 + y^2 + z^2 = p^2 [\sin^2 A (1) + \cos^2 A]\)
\(x^2 + y^2 + z^2 = p^2 [\sin^2 A + \cos^2 A]\)
Now, we can use the same trigonometric identity again, but this time for angle A: \(\sin^2 A + \cos^2 A = 1\).
Substitute this into the expression:
\(x^2 + y^2 + z^2 = p^2 [1]\)
\(x^2 + y^2 + z^2 = p^2\)
Thus, the value of \(x^2 + y^2 + z^2\) is \(p^2\).
Revision Table: Key Concepts
Review the key elements used in solving this problem.
| Concept | Description | Application in Problem |
|---|---|---|
| Squaring expressions | Multiplying a term by itself. \((ab)^2 = a^2b^2\) | Used to find \(x^2\), \(y^2\), and \(z^2\) |
| Factoring | Extracting a common factor from terms. | Factored out \(p^2\) and \(\sin^2 A\) |
| Trigonometric Identity | \(\sin^2 \theta + \cos^2 \theta = 1\) | Used twice for angles B and A |
Additional Information: Fundamental Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variables involved (where they are defined). The identity used in this problem, \(\sin^2 \theta + \cos^2 \theta = 1\), is one of the most fundamental Pythagorean identities.
Understanding and applying these identities is crucial for simplifying trigonometric expressions and solving related problems. This identity comes directly from the Pythagorean theorem applied to a right-angled triangle inscribed in a unit circle, where the hypotenuse is 1, the opposite side is \(\sin \theta\), and the adjacent side is \(\cos \theta\).
Other basic identities include reciprocal identities (like \(\sec \theta = \frac{1}{\cos \theta}\)) and quotient identities (like \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)). Mastering these basic identities provides a strong foundation for more advanced trigonometry.
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