Given that θ is measured in radians, prove, from first principles, that the derivative of sin θ is cos θ - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 1
Question 11
Given that θ is measured in radians, prove, from first principles, that the derivative of sin θ is cos θ.
You may assume the formula for sin(A ± B) and that as h → 0... show full transcript
Worked Solution & Example Answer:Given that θ is measured in radians, prove, from first principles, that the derivative of sin θ is cos θ - Edexcel - A-Level Maths Pure - Question 11 - 2017 - Paper 1
Step 1
Use of \( ext{sin}( heta + h) - ext{sin}( heta) \)
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Answer
Start with the definition of the derivative:
hsin(θ+h)−sin(θ)
Using the compound angle identity, we can write:
sin(θ+h)=sin(θ)cos(h)+cos(θ)sin(h)
This gives us:
hsin(θ)cos(h)+cos(θ)sin(h)−sin(θ)
Step 2
Simplifying the expression
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Answer
We can factor out the terms:
hsin(θ)(cos(h)−1)+cos(θ)sin(h)
This can be split into:
sin(θ)hcos(h)−1+cos(θ)hsin(h)
Step 3
Taking the limit as h approaches 0
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Answer
As ( h \to 0 ):
( \frac{\text{sin}(h)}{h} \to 1 )
( \frac{\text{cos}(h) - 1}{h} \to 0 )
Thus, the limit can be computed as:
h→0lim(sin(θ)⋅0+cos(θ)⋅1)=cos(θ)
Step 4
Conclusion
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