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Sagot :
To determine the minimum value of the expression [tex]\(\sin^4 \theta + \cos^4 \theta\)[/tex], we can use trigonometric identities and some algebraic manipulation. Follow these steps to solve the problem:
1. Use the Pythagorean Identity:
Recall the fundamental trigonometric identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
2. Rewrite the expression:
Let [tex]\( x = \sin^2 \theta \)[/tex]. Then, [tex]\( \cos^2 \theta = 1 - x \)[/tex]. Substituting these into our original expression, we get:
[tex]\[ \sin^4 \theta + \cos^4 \theta = x^2 + (1 - x)^2 \][/tex]
3. Simplify the new expression:
Expand and simplify the expression:
[tex]\[ x^2 + (1 - x)^2 = x^2 + 1 - 2x + x^2 = 2x^2 - 2x + 1 \][/tex]
4. Recognize the form:
Notice that [tex]\( 2x^2 - 2x + 1 \)[/tex] is a quadratic function in terms of [tex]\( x \)[/tex]. A quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex] reaches its minimum value at the vertex, given by [tex]\( x = -\frac{b}{2a} \)[/tex].
5. Find the vertex:
For the quadratic function [tex]\( 2x^2 - 2x + 1 \)[/tex], we have [tex]\( a = 2 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 1 \)[/tex]. The vertex [tex]\( x \)[/tex]-coordinate is:
[tex]\[ x = -\frac{-2}{2 \cdot 2} = \frac{2}{4} = \frac{1}{2} \][/tex]
6. Calculate the minimum value:
Substitute [tex]\( x = \frac{1}{2} \)[/tex] back into the function [tex]\( 2x^2 - 2x + 1 \)[/tex]:
[tex]\[ f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) + 1 \][/tex]
[tex]\[ = 2 \cdot \frac{1}{4} - 2 \cdot \frac{1}{2} + 1 \][/tex]
[tex]\[ = \frac{1}{2} - 1 + 1 \][/tex]
[tex]\[ = \frac{1}{2} \][/tex]
Therefore, the minimum value of [tex]\(\sin^4 \theta + \cos^4 \theta\)[/tex] is [tex]\(\frac{1}{2}\)[/tex]. The correct answer is:
a. [tex]\(\frac{1}{2}\)[/tex]
1. Use the Pythagorean Identity:
Recall the fundamental trigonometric identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
2. Rewrite the expression:
Let [tex]\( x = \sin^2 \theta \)[/tex]. Then, [tex]\( \cos^2 \theta = 1 - x \)[/tex]. Substituting these into our original expression, we get:
[tex]\[ \sin^4 \theta + \cos^4 \theta = x^2 + (1 - x)^2 \][/tex]
3. Simplify the new expression:
Expand and simplify the expression:
[tex]\[ x^2 + (1 - x)^2 = x^2 + 1 - 2x + x^2 = 2x^2 - 2x + 1 \][/tex]
4. Recognize the form:
Notice that [tex]\( 2x^2 - 2x + 1 \)[/tex] is a quadratic function in terms of [tex]\( x \)[/tex]. A quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex] reaches its minimum value at the vertex, given by [tex]\( x = -\frac{b}{2a} \)[/tex].
5. Find the vertex:
For the quadratic function [tex]\( 2x^2 - 2x + 1 \)[/tex], we have [tex]\( a = 2 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 1 \)[/tex]. The vertex [tex]\( x \)[/tex]-coordinate is:
[tex]\[ x = -\frac{-2}{2 \cdot 2} = \frac{2}{4} = \frac{1}{2} \][/tex]
6. Calculate the minimum value:
Substitute [tex]\( x = \frac{1}{2} \)[/tex] back into the function [tex]\( 2x^2 - 2x + 1 \)[/tex]:
[tex]\[ f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) + 1 \][/tex]
[tex]\[ = 2 \cdot \frac{1}{4} - 2 \cdot \frac{1}{2} + 1 \][/tex]
[tex]\[ = \frac{1}{2} - 1 + 1 \][/tex]
[tex]\[ = \frac{1}{2} \][/tex]
Therefore, the minimum value of [tex]\(\sin^4 \theta + \cos^4 \theta\)[/tex] is [tex]\(\frac{1}{2}\)[/tex]. The correct answer is:
a. [tex]\(\frac{1}{2}\)[/tex]
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