Back to the full dot-point answer

NSWMaths Extension 1Quick questions

Trigonometric Functions (ME-T1, T2, T3)

Quick questions on Product-to-sum and sum-to-product identities for trigonometric expressions

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 four product-to-sum identities?
Show answer
Derived by adding or subtracting the sum and difference identities:
What is derivation?
Show answer
Add sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A \cos B + \cos A \sin B and sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A \cos B - \cos A \sin B. The cos⁡Asin⁡B\cos A \sin B terms cancel: sin⁡(A+B)+sin⁡(A−B)=2sin⁡Acos⁡B\sin(A + B) + \sin(A - B) = 2 \sin A \cos B. Divide by 22 to get the first identity.
What is sum-to-product (the converses)?
Show answer
Let A+B=PA + B = P and A−B=QA - B = Q, so A=P+Q2A = \frac{P + Q}{2} and B=P−Q2B = \frac{P - Q}{2}. Substituting:
What is why these identities matter?
Show answer
Three main uses appear in HSC questions.
What is integrate a product?
Show answer
Evaluate ∫2sin⁡3xcos⁡x dx\int 2 \sin 3 x \cos x \, dx.
What is sum to product?
Show answer
Simplify sin⁡5x+sin⁡3x\sin 5 x + \sin 3 x.
What is product of two cosines?
Show answer
Express cos⁡4xcos⁡2x\cos 4 x \cos 2 x as a sum.
What is mixing up product-to-sum signs?
Show answer
sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A \sin B = \tfrac{1}{2}[\cos(A - B) - \cos(A + B)] is the difference of cosines, in that order. Get the order wrong and you flip the sign.
What is using A−BA - B when A<BA < B?
Show answer
sin⁡Acos⁡B\sin A \cos B uses A−BA - B even if that is negative. sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta handles it cleanly.
What is forgetting the factor of 12\tfrac{1}{2}?
Show answer
Every product-to-sum identity has a 12\tfrac{1}{2} out the front; every sum-to-product has a 22.
What is algebra inside the identity?
Show answer
A+BA + B and A−BA - B must be evaluated in radians or degrees consistently. Mix the two and the answer is garbage. :::

All Maths Extension 1Q&A pages