Range of Expressions using Trigonometric Substitution
Range of different trigonometric functions helps us to use these functions as substitution in many algebraic expressions.
This simplifies the give expression to another expression, the range of which can be easily determined.
Example 1: If x² + y² = 4, then find the maximum value of \(\frac{{{x}^{3}}+{{y}^{3}}}{x+y}\).
Solution: Given that x² + y² = 4
Let us consider x = 2cosθ, y = 2sinθ
Now, \(\frac{{{x}^{3}}+{{y}^{3}}}{x+y}={{x}^{2}}+{{y}^{2}}-xy\),
= 4cos²θ + 4sin²θ – (2sinθ) (2cosθ)
= 4(cos²θ + sin²θ) – 4 sinθ cosθ
= 4(1) – 2sin2θ
\({{\left( \frac{{{x}^{3}}+{{y}^{3}}}{x+y} \right)}_{\max }}\) = 4 (1) – 2 (-1) = 6
Example 2: If \(\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\), then find the range of 2x + y.
Solution: Given that \(\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{9}=1\),
\({{\left( \frac{x}{2} \right)}^{2}}+{{\left( \frac{y}{3} \right)}^{2}}=1\)Let us consider x = 2cosθ, y = 3sinθ
2x + y = 2(2cosθ) + 3sinθ
(4cosθ) + 3sinθ
\(a\cos \theta +b\sin \theta \in \left[ -\sqrt{{{a}^{2}}+{{b}^{2}}},\sqrt{{{a}^{3}}+{{b}^{2}}} \right]\),
\(4\cos \theta +3\sin \theta \in \left[ -\sqrt{{{4}^{2}}+{{3}^{2}}},\sqrt{{{4}^{3}}+{{3}^{2}}} \right]\),
4 cosθ + 3 sinθ ϵ [-5, 5]
2x + y ϵ [-5, 5].
Example 3: if x² + y² = x²y², then find the range of .
Solution: Given that x² + y² = x²y²
Let us consider x = secθ and y = cosecθ
\(\frac{5x+12y+7xy}{xy}\),
\(=\frac{5\sec \theta +12\cos ec\theta +7\sec \theta \times \cos ec\theta }{sec\theta \times \cos ec\theta }\),
\(=\frac{5\frac{1}{\cos \theta }+12\frac{1}{\sin \theta }+7\frac{1}{\cos \theta }\times \frac{1}{\sin \theta }}{\frac{1}{\sin \theta }\times \frac{1}{\cos \theta }}\),
= 5 sinθ + 12 cosθ + 7
\(5\sin \theta +12\cos \theta +7\in \left[ -\sqrt{{{5}^{2}}+{{12}^{2}}-7},\sqrt{{{5}^{2}}+{{12}^{2}}+7} \right]\),
5 sinθ + 12 cosθ + 7 ϵ [-6, 20],
\(\frac{5x+12y+7xy}{xy}\in \left[ -6,20 \right]\).