When $\theta$ is small, find an approximation for $\cos 30^\circ + \theta \sin 2\theta$, giving your answer in the form $a + b\theta^2$. - AQA - A-Level Maths Mechanics - Question 3 - 2017 - Paper 1
Question 3
When $\theta$ is small, find an approximation for $\cos 30^\circ + \theta \sin 2\theta$, giving your answer in the form $a + b\theta^2$.
Worked Solution & Example Answer:When $\theta$ is small, find an approximation for $\cos 30^\circ + \theta \sin 2\theta$, giving your answer in the form $a + b\theta^2$. - AQA - A-Level Maths Mechanics - Question 3 - 2017 - Paper 1
Step 1
Use approximations for trigonometric functions
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
For small angles, we can use the approximations:
cosx≈1−2x2
sinx≈x
Thus, for θ, we approximate:
sin2θ≈2θ
cos30∘=23 (exact, as 30∘ is not small).
Step 2
Substitute $2\theta$ and $30^\circ$ into the expression
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
We can rewrite the expression as follows: cos30∘+θsin2θ≈23+θ(2θ).
Thus, we have: cos30∘+θsin2θ≈23+2θ2.
Step 3
Obtain the correct answer in the form $a + b\theta^2$
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
This expression simplifies to: 23+2θ2.
In this case, a=23 and b=2, therefore our final answer is in the required form.
