4. (a) Express $6 \, \cos \theta + 8 \, \sin \theta$ in the form $R \cos (\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 25 - 2013 - Paper 1
Question 25
4. (a) Express $6 \, \cos \theta + 8 \, \sin \theta$ in the form $R \cos (\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$.
Give the value of $\al... show full transcript
Worked Solution & Example Answer:4. (a) Express $6 \, \cos \theta + 8 \, \sin \theta$ in the form $R \cos (\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 25 - 2013 - Paper 1
Step 1
Express $6 \cos \theta + 8 \sin \theta$ in the form $R \cos (\theta - \alpha)$
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
First, we calculate R using Pythagoras' theorem:
R2=62+82=36+64=100⇒R=10.
Next, we find α using the tangent function:
tanα=68⇒α=tan−1(68)≈0.927 radians.
Therefore, the expression is given by:
6cosθ+8sinθ=10cos(θ−0.927).
Step 2
Calculate the maximum value of $p(\theta)$
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
To find the maximum value of p(θ):
Given:
p(θ)=12+6cosθ+8sinθ4.
The minimum value of the denominator occurs when 6cosθ+8sinθ is maximized, which was previously determined to be 10. Therefore:
12+10=22
Thus:
Maximum of p(θ)=224=0.181.
Step 3
Determine the value of $\theta$ at which the maximum occurs
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
The maximum occurs when:
6cosθ+8sinθ=10
Thus:
cos(θ−0.927)=1⇒θ−0.927=0⇒θ=0.927.
