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NSWMaths Extension 1Quick questions

Trigonometric Functions (ME-T1, T2, T3)

Quick questions on Sum and difference identities for sin, cos and tan: expansions, simplifications and exact values

11short 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 sum and difference identities?
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$sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B,\sin(A + B) = \sin A \cos B + \cos A \sin B, sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B.\sin(A - B) = \sin A \cos B - \cos A \sin B.$
What is derivations of double-angle from sum identities?
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$sin⁡2θ=sin⁡(θ+θ)=2sin⁡θcos⁡θ.\sin 2\theta = \sin(\theta + \theta) = 2 \sin \theta \cos \theta.$
What is half-angle identities?
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From the double-angle forms of cos⁡2θ\cos 2\theta,
What is exact values for non-standard angles?
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The standard angles 0,30,45,60,90∘0, 30, 45, 60, 90^\circ have well-known exact values. Combining them with sum and difference identities gives exact values for 15,75,105,165∘15, 75, 105, 165^\circ and so on.
What is simplify a sum?
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Simplify sin⁡(θ+π4)+sin⁡(θ−π4)\sin(\theta + \frac{\pi}{4}) + \sin(\theta - \frac{\pi}{4}).
What is prove an identity?
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Prove cos⁡(θ+60∘)+cos⁡(θ−60∘)=cos⁡θ\cos(\theta + 60^\circ) + \cos(\theta - 60^\circ) = \cos \theta.
What is combine with double-angle?
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Given sin⁡θ=35\sin \theta = \frac{3}{5} in Q1, find sin⁡3θ\sin 3\theta.
What is wrong quadrant for cos⁡β\cos \beta or sin⁡β\sin \beta?
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When the question specifies a quadrant, get the sign of the other trig ratio right before substituting.
What is forgetting to rationalise?
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A final answer of 13−1\frac{1}{\sqrt{3} - 1} should be rationalised to 3+12\frac{\sqrt{3} + 1}{2}. Markers expect simplified surds.
What is tangent sum with zero denominator?
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tan⁡(45∘+45∘)\tan(45^\circ + 45^\circ) would have 1−1⋅1=01 - 1 \cdot 1 = 0 in the denominator; the identity is undefined and you must reason via sin⁡/cos⁡\sin/\cos directly.
What is reusing the wrong angle?
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sin⁡105∘=sin⁡(60+45)∘\sin 105^\circ = \sin(60 + 45)^\circ, not sin⁡(75+30)∘\sin(75 + 30)^\circ in the most efficient form. Choose decompositions into standard angles. :::

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