Prove that √2 × sin(π/4 - θ) equals cos(θ) - sin(θ).
Understand the Problem
The question is asking to prove a trigonometric identity involving square roots and sine and cosine functions. The identity states that √2 × sin(π/4 - θ) is equal to cos(θ) - sin(θ). This will involve using trigonometric identities and properties of sine and cosine functions to show both sides of the equation are equivalent.
Answer
The identity is proven: $$ \sqrt{2} \sin\left(\frac{\pi}{4} - \theta\right) = \cos(\theta) - \sin(\theta) $$
Answer for screen readers
The identity is proven:
$$ \sqrt{2} \sin\left(\frac{\pi}{4} - \theta\right) = \cos(\theta) - \sin(\theta) $$
Steps to Solve
- Apply the sine subtraction formula
We will use the sine subtraction formula, which states that:
$$ \sin(a - b) = \sin(a) \cos(b) - \cos(a) \sin(b) $$
In our case, let $a = \frac{\pi}{4}$ and $b = \theta$. Then:
$$ \sin\left(\frac{\pi}{4} - \theta\right) = \sin\left(\frac{\pi}{4}\right) \cos(\theta) - \cos\left(\frac{\pi}{4}\right) \sin(\theta) $$
Since we know:
$$ \sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
We can substitute these values into the equation:
$$ \sin\left(\frac{\pi}{4} - \theta\right) = \frac{\sqrt{2}}{2} \cos(\theta) - \frac{\sqrt{2}}{2} \sin(\theta) $$
- Multiply through by $\sqrt{2}$
Now, we multiply both sides of the equation by $\sqrt{2}$:
$$ \sqrt{2} \sin\left(\frac{\pi}{4} - \theta\right) = \sqrt{2} \left(\frac{\sqrt{2}}{2} \cos(\theta) - \frac{\sqrt{2}}{2} \sin(\theta)\right) $$
This simplifies to:
$$ \sqrt{2} \sin\left(\frac{\pi}{4} - \theta\right) = \cos(\theta) - \sin(\theta) $$
- Complete the proof
Now we have shown that:
$$ \sqrt{2} \sin\left(\frac{\pi}{4} - \theta\right) = \cos(\theta) - \sin(\theta) $$
This completes our proof that the two sides of the equation are equal.
The identity is proven:
$$ \sqrt{2} \sin\left(\frac{\pi}{4} - \theta\right) = \cos(\theta) - \sin(\theta) $$
More Information
This proof shows that the left side simplifies to the right side using basic trigonometric identities and properties. Trigonometric identities are essential tools in mathematics and are used in various fields such as physics, engineering, and computer science.
Tips
- Forgetting to apply the sine subtraction formula correctly.
- Not substituting the values of sine and cosine at specific angles properly.
- Failing to multiply both sides of the equation after applying the formula.
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