What is the least value of 3sin 2 θ + 4cos 2 θ?
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Finding the Least Value of Trigonometric Expressions
We are asked to find the least value of the expression \(3\sin^2\theta + 4\cos^2\theta\). To find the minimum or maximum value of a trigonometric expression, we often try to rewrite it in a simpler form using trigonometric identities.
Using Trigonometric Identities to Simplify
The fundamental trigonometric identity is \(\sin^2\theta + \cos^2\theta = 1\). We can use this identity to express one term in terms of the other.
Let the given expression be \(E\).
\(E = 3\sin^2\theta + 4\cos^2\theta\)
We can rewrite \(4\cos^2\theta\) as \((3+1)\cos^2\theta\) or split the expression:
\(E = 3\sin^2\theta + 3\cos^2\theta + \cos^2\theta\)
Now, factor out 3 from the first two terms:
\(E = 3(\sin^2\theta + \cos^2\theta) + \cos^2\theta\)
Using the identity \(\sin^2\theta + \cos^2\theta = 1\), we substitute this into the expression:
\(E = 3(1) + \cos^2\theta\)
\(E = 3 + \cos^2\theta\)
Determining the Range of \(\cos^2\theta\)
To find the least value of \(3 + \cos^2\theta\), we need to know the possible values of \(\cos^2\theta\).
We know that the range of \(\cos\theta\) for any real value of \(\theta\) is \([-1, 1]\).
\(-1 \le \cos\theta \le 1\)
When we square \(\cos\theta\), the values in the range \([-1, 1]\) become non-negative. The smallest value is \(0^2 = 0\) (when \(\cos\theta = 0\)), and the largest value is \((-1)^2 = 1\) or \(1^2 = 1\) (when \(\cos\theta = -1\) or \(\cos\theta = 1\)).
So, the range of \(\cos^2\theta\) is \([0, 1]\).
\(0 \le \cos^2\theta \le 1\)
Finding the Least Value of the Expression
The expression is \(E = 3 + \cos^2\theta\). To find the least value of \(E\), we must use the least value of \(\cos^2\theta\). The least value of \(\cos^2\theta\) is 0.
Substitute the minimum value of \(\cos^2\theta\) into the expression:
Least Value of \(E = 3 + (\text{Least Value of } \cos^2\theta)\)
Least Value of \(E = 3 + 0\)
Least Value of \(E = 3\)
Alternatively, we could have rewritten the expression as \(E = 4 - \sin^2\theta\). The range of \(\sin^2\theta\) is also \([0, 1]\). To get the least value of \(4 - \sin^2\theta\), we need to subtract the largest possible value of \(\sin^2\theta\), which is 1.
Least Value of \(E = 4 - (\text{Largest Value of } \sin^2\theta)\)
Least Value of \(E = 4 - 1\)
Least Value of \(E = 3\)
Both methods give the same result, confirming the least value is 3.
Summary of the Calculation
| Step | Description | Expression |
|---|---|---|
| 1 | Start with the given expression | \(3\sin^2\theta + 4\cos^2\theta\) |
| 2 | Rewrite using \(\cos^2\theta = 1 - \sin^2\theta\) or \(\sin^2\theta = 1 - \cos^2\theta\) | \(3\sin^2\theta + 4(1 - \sin^2\theta)\) OR \(3(1 - \cos^2\theta) + 4\cos^2\theta\) |
| 3 | Simplify the expression | \(4 - \sin^2\theta\) OR \(3 + \cos^2\theta\) |
| 4 | Determine the range of \(\sin^2\theta\) or \(\cos^2\theta\) | \(0 \le \sin^2\theta \le 1\) OR \(0 \le \cos^2\theta \le 1\) |
| 5 | Find the least value using the range | \(4 - (\text{max value of }\sin^2\theta) = 4 - 1 = 3\) OR \(3 + (\text{min value of }\cos^2\theta) = 3 + 0 = 3\) |
The least value of \(3\sin^2\theta + 4\cos^2\theta\) is 3.
Revision Table: Key Trigonometric Concepts
| Concept | Description | Identity/Range |
|---|---|---|
| Trigonometric Identity | Fundamental relationship between \(\sin\theta\) and \(\cos\theta\). | \(\sin^2\theta + \cos^2\theta = 1\) |
| Range of \(\cos\theta\) | All possible output values for \(\cos\theta\). | \([-1, 1]\) |
| Range of \(\sin\theta\) | All possible output values for \(\sin\theta\). | \([-1, 1]\) |
| Range of \(\cos^2\theta\) | All possible output values for \(\cos^2\theta\). | \([0, 1]\) |
| Range of \(\sin^2\theta\) | All possible output values for \(\sin^2\theta\). | \([0, 1]\) |
Additional Information on Finding Extrema
When asked to find the minimum or maximum value (extrema) of a trigonometric expression involving powers of \(\sin\theta\) and \(\cos\theta\), especially \(\sin^2\theta\) and \(\cos^2\theta\), a common strategy is to use the identity \(\sin^2\theta + \cos^2\theta = 1\) to reduce the expression to a form involving only one squared trigonometric function, plus possibly a constant.
For an expression of the form \(a\sin^2\theta + b\cos^2\theta\), we can rewrite it as:
- \(a\sin^2\theta + b(1 - \sin^2\theta) = a\sin^2\theta + b - b\sin^2\theta = (a-b)\sin^2\theta + b\)
- \(a(1 - \cos^2\theta) + b\cos^2\theta = a - a\cos^2\theta + b\cos^2\theta = a + (b-a)\cos^2\theta\)
In our case, \(a=3\) and \(b=4\). Using the second form: \(3 + (4-3)\cos^2\theta = 3 + \cos^2\theta\).
Since the range of \(\cos^2\theta\) is \([0, 1]\):
- The minimum value occurs when \(\cos^2\theta = 0\), giving \(3+0 = 3\).
- The maximum value occurs when \(\cos^2\theta = 1\), giving \(3+1 = 4\).
So the range of the expression \(3\sin^2\theta + 4\cos^2\theta\) is \([3, 4]\). The least value is indeed 3.
This method is effective for expressions that can be simplified using \(\sin^2\theta + \cos^2\theta = 1\). Other techniques might be needed for different types of trigonometric expressions.
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