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Albert1
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$0<\theta <90^o$
using geometry to prove the following:
$1<sin\,\, \theta + cos\,\,\theta \leq \sqrt 2$
using geometry to prove the following:
$1<sin\,\, \theta + cos\,\,\theta \leq \sqrt 2$
my soluionAlbert said:$0<\theta <90^o$
using geometry to prove the following:
$1<sin\,\, \theta + cos\,\,\theta \leq \sqrt 2$
The inequality $1 The inequality is only true for angles between 0 and 90 degrees because the sine and cosine values fall within this range. Beyond 90 degrees, the sine and cosine values become negative, and the inequality would no longer hold true. We can prove this inequality using the Pythagorean identity, which states that $sin^2\,\theta + cos^2\,\theta = 1$. By rearranging this identity, we get $sin\,\theta + cos\,\theta = \sqrt{1-sin^2\,\theta}$. Since $sin^2\,\theta$ is always less than or equal to 1 for angles between 0 and 90 degrees, we can substitute this value into the equation to get $sin\,\theta + cos\,\theta \leq \sqrt 2$. We can also see that the minimum value for $sin\,\theta + cos\,\theta$ is achieved when $sin^2\,\theta = 0$, which results in $sin\,\theta + cos\,\theta = 1$. Therefore, the inequality $1 The inequality $1 Yes, this inequality can be applied to other trigonometric functions, such as tangent and secant, by using their respective identities and manipulating the equations. However, the range of angles for which the inequality holds true may differ for each function.2. Why is the inequality only true for angles between 0 and 90 degrees?
3. How can we prove that $1
4. What is the significance of the inequality $1
5. Can this inequality be applied to other trigonometric functions?
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