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NSWMaths Extension 1Quick questions

Trigonometric Functions (ME-T1, T2, T3)

Quick questions on Auxiliary angle: writing asin⁡x+bcos⁡xa \sin x + b \cos x as Rsin⁡(x+α)R \sin(x + \alpha)

10short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the form Rsin⁡(x+α)R \sin(x + \alpha)?
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Expand using the sine sum identity:
What is other equivalent forms?
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Depending on convenience, asin⁡x+bcos⁡xa \sin x + b \cos x can also be written as:
What is why this matters?
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Once asin⁡x+bcos⁡xa \sin x + b \cos x is written as Rsin⁡(x+α)R \sin(x + \alpha):
What is equation?
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Solve sin⁡x+3cos⁡x=1\sin x + \sqrt{3} \cos x = 1 for 0≤x≤2π0 \le x \le 2 \pi.
What is maximum and minimum?
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Find the maximum value of 5sin⁡x−12cos⁡x5 \sin x - 12 \cos x and the smallest positive xx at which it occurs.
What is sketching?
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R=2R = \sqrt{2}, α=π4\alpha = \frac{\pi}{4}, so y=2sin⁡ ⁣(x+π4)y = \sqrt{2} \sin\!\left( x + \frac{\pi}{4} \right).
What is wrong quadrant for α\alpha?
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tan⁡α=ba\tan \alpha = \frac{b}{a} has two candidate angles per period. Choose the one consistent with the signs of Rcos⁡α=aR \cos \alpha = a and Rsin⁡α=bR \sin \alpha = b.
What is forgetting to halve when solving the equation?
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Rsin⁡(x+α)=cR \sin(x + \alpha) = c requires dividing by RR, then taking arcsin⁡\arcsin, then accounting for both branches.
What is confusing amplitude with max/min?
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Maximum value of Rsin⁡(x+α)+cR \sin(x + \alpha) + c is R+cR + c, minimum is c−Rc - R, amplitude is RR. Track each.
What is negative RR?
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Convention is R>0R > 0. If you write −Rsin⁡(x+α)=Rsin⁡(x+α+π)-R \sin(x + \alpha) = R \sin(x + \alpha + \pi), you can keep RR positive by shifting α\alpha by π\pi. :::

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