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Some students are trying to prove an identity for sin(A + B) - AQA - A-Level Maths Mechanics - Question 14 - 2018 - Paper 1
Question 14
Some students are trying to prove an identity for sin(A + B). They start by drawing two right-angled triangles ODE and OEF, as shown. The students' incomplete proof... show full transcript
Worked Solution & Example Answer:Some students are trying to prove an identity for sin(A + B) - AQA - A-Level Maths Mechanics - Question 14 - 2018 - Paper 1
Step 1
Explain why PF / EF × EF / OF leads to cos A sin B in Line 5.
Answer
In the proof, the expression PF / EF represents the adjacent side over the hypotenuse in the triangle EOF, which by the definition of cosine corresponds to cos A. Similarly, the expression EF / OF is the ratio of the opposite side to the hypotenuse in the triangle OFR, corresponding to sin B. Therefore, by multiplying these two ratios, we derive the term cos A sin B as shown in Line 5.
Step 3
Explain why the argument used in part (a) only proves the identity when A and B are acute angles.
Answer
The argument in part (a) relies on the definitions and properties of sine and cosine for acute angles where both functions have positive values. If A and B are not acute, the values of sine and cosine may take negative forms or become undefined, invalidating the proof.
Step 4
Another student claims that by replacing B with -B in the identity for sin(A + B) it is possible to find an identity for sin(A - B).
Answer
Starting with the identity for sin(A + B):
Substituting -B in place of B gives:
Using the even and odd properties of sine and cosine, we can simplify it to:
Thus, we derive the identity for sin(A - B).