How to write 2sinθ - 4cosθ in the form rsin(θ - α)?
How to write 2sinθ - 4cosθ in the form rsin(θ - α)?
Hence solve 2 2sinθ - 4cosθ = 3 for 0≤ θ ≤360.
How to write 2sinθ - 4cosθ in the form rsin(θ - α)?
Hence solve 2 2sinθ - 4cosθ = 3 for 0≤ θ ≤360.
Part 1:
Part 2:
Equating parts,
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To write (2\sin\theta - 4\cos\theta) in the form (r\sin(\theta - \alpha)), you can use the following steps:
- Use the trigonometric identity: (r\sin(\theta - \alpha) = r(\sin\theta\cos\alpha - \cos\theta\sin\alpha)).
- Compare the expression (2\sin\theta - 4\cos\theta) with (r(\sin\theta\cos\alpha - \cos\theta\sin\alpha)).
- Equate the coefficients of (\sin\theta) and (\cos\theta) in both expressions to find (r) and (\alpha).
Let's proceed with the calculations:
Given expression: (2\sin\theta - 4\cos\theta)
Comparing with (r(\sin\theta\cos\alpha - \cos\theta\sin\alpha)):
Coefficient of (\sin\theta): (2 = r\cos\alpha)
Coefficient of (\cos\theta): (-4 = -r\sin\alpha)
From the first equation, solve for (r): (r = \frac{2}{\cos\alpha})
From the second equation, solve for (\alpha): (\sin\alpha = \frac{4}{r} = \frac{4}{\frac{2}{\cos\alpha}} = \frac{4\cos\alpha}{2} = 2\cos\alpha)
Now, square both sides: (\sin^2\alpha = 4\cos^2\alpha)
Use the identity (\sin^2\alpha + \cos^2\alpha = 1): (1 - \cos^2\alpha = 4\cos^2\alpha)
Rearrange and solve for (\cos^2\alpha): (5\cos^2\alpha = 1) (\cos^2\alpha = \frac{1}{5})
Take the square root: (\cos\alpha = \pm \frac{1}{\sqrt{5}})
Since (r = \frac{2}{\cos\alpha}), and (\cos\alpha) cannot be negative for (0 < \theta < 360^\circ), we take the positive value for (\cos\alpha).
(\cos\alpha = \frac{1}{\sqrt{5}})
Now, substitute the value of (r) into the equation: (r = \frac{2}{\frac{1}{\sqrt{5}}} = 2\sqrt{5})
So, (r = 2\sqrt{5}) and (\alpha = \arccos\left(\frac{1}{\sqrt{5}}\right)).
Therefore, (2\sin\theta - 4\cos\theta = 2\sqrt{5}\sin\left(\theta - \arccos\left(\frac{1}{\sqrt{5}}\right)\right)).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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