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NSWMaths Extension 1Quick questions

Trigonometric Functions (ME-T1, T2, T3)

Quick questions on The t-formula: rational expressions for sin⁡θ\sin \theta, cos⁡θ\cos \theta and tan⁡θ\tan \theta via t=tan⁡(θ/2)t = \tan(\theta/2)

12short 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 substitution?
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Let t=tan⁡θ2t = \tan \frac{\theta}{2}. Then
What is derivation?
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Let ϕ=θ2\phi = \frac{\theta}{2}, so θ=2ϕ\theta = 2 \phi and t=tan⁡ϕt = \tan \phi.
What is when to use the t-formula?
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The t-formula is most useful for:
What is limitations?
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The substitution t=tan⁡θ2t = \tan \frac{\theta}{2} is undefined at θ=π+2nπ\theta = \pi + 2 n \pi (where cos⁡θ2=0\cos \frac{\theta}{2} = 0).
What is algebraic conversion?
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To solve asin⁡θ+bcos⁡θ=ca \sin \theta + b \cos \theta = c with t-formula:
What is verify the identity?
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Show 1−cos⁡θsin⁡θ=tan⁡θ2\frac{1 - \cos \theta}{\sin \theta} = \tan \frac{\theta}{2}.
What is express a sum?
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Express sin⁡θ+cos⁡θ\sin \theta + \cos \theta in terms of t=tan⁡θ2t = \tan \frac{\theta}{2}.
What is integral via t-formula?
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Evaluate ∫dθ1+sin⁡θ\int \frac{d \theta}{1 + \sin \theta}.
What is sign error after 1−t21 - t^2?
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cos⁡θ=1−t21+t2\cos \theta = \frac{1 - t^2}{1 + t^2} is positive when ∣t∣<1|t| < 1 and negative when ∣t∣>1|t| > 1. Sketch a quick number line if unsure.
What is treating tt as the unknown?
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Solve for tt first, then recover θ=2arctan⁡t\theta = 2 \arctan t at the end. The general solution for θ\theta may include extra branches.
What is algebra failures when cross-multiplying?
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2t1+t2\frac{2 t}{1 + t^2} multiplied by (1+t2)(1 + t^2) becomes 2t2 t, not 22. Track the cancellation.
What is confusing tan⁡\tan with 12tan⁡θ\frac{1}{2} \tan \theta?
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These are different functions. The t-formula uses the former. :::

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