Aussie Maths & Science Teachers: Save your time with SmarterEd
For real numbers \(a\) and \(b\), where \(a \neq 0\) and \(b \neq 0\), we can find numbers \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) and \(R\) such that \(a\,\cos x + b\,\sin x\) can be written in the following 4 forms:
\(R\,\sin(x + \alpha)\)
\(R\,\sin(x-\beta)\)
\(R\,\cos(x + \gamma)\)
\(R\,\cos(x-\delta)\)
where \(R \gt 0\) and \(0<\alpha, \beta, \gamma, \delta \lt 2\pi\).
What is the value of \(\alpha + \beta + \gamma + \delta\)?
\(D\)
\(a\,\sin\,x+b\,\cos\,x\ \ \text{can be written 4 ways.}\)
\(\text{Consider the case:}\)
\(\cos\,x+\sin\,x\ \ \text{where}\ a=b=1,\ \ R=\sqrt{2}\)
| \(\sqrt{2}\,\sin(x+\alpha)\) | \(= \sqrt{2}(\sin\,x\,\cos\,\alpha + \cos\,x\,\sin\,\alpha)\) | |
| \(= \sqrt{2}\Big(\sin\,x \cdot \dfrac{1}{\sqrt2} + \cos\,x \cdot \dfrac{1}{\sqrt2}\Big)\) |
\(\cos\, \alpha^{+}, \sin\,\alpha^{+}\ \Rightarrow\ \ \alpha=\dfrac{\pi}{4} \)
\(\text{Similarly for other 3 cases:}\)
\(\sqrt{2}\,\sin(x-\beta): \ \cos\, \beta^{+}, \sin\,\beta^{-}\ \Rightarrow\ \ \beta=\dfrac{7\pi}{4} \)
\(\sqrt{2}\,\cos(x+\gamma): \ \cos\, \gamma^{+}, \sin\,\gamma^{-}\ \Rightarrow\ \ \gamma=\dfrac{7\pi}{4} \)
\(\sqrt{2}\,\cos(x-\delta): \ \cos\, \delta^{+}, \sin\,\delta^{+}\ \Rightarrow\ \ \delta=\dfrac{\pi}{4} \)
\(\therefore \alpha + \beta + \gamma + \delta = 4\pi\)
\(\Rightarrow D\)
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Give your answers correct to three decimal places. (2 marks)
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i. `text(Write)\ \ 8 cos x + 6 sin x\ \ text(in the form)\ \ A cos (x − α)`
| `A(cos\ x\ cos\ α + sin\ x\ sin\ α)` | `= 8\ cos\ x + 6\ sin\ x` |
| `(cos\ x\ cos\ α + sin\ x\ sin\ α)` | `= 8/A\ cos\ x + 6/A\ sin\ x` |
`⇒ cos\ α = 8/A`
`⇒ sin\ α = 6/A`
| `(8/A)^2 + (6/A)^2` | `= 1` |
| `8^2 + 6^2` | `= A^2` |
| `:.A` | `= sqrt100` |
| `= 10` |
| `=>cos\ α` | `= 8/10` |
| `:.α` | `= 0.6435…\ text(radians)` |
`:.8 cos x + 6 sin x = 10 cos (x − 0.6435…)`
| ii. | `8\ cos\ x + 6\ sin\ x` | `= 5` |
| `10\ cos\ (x − 0.6435…)` | `= 5` | |
| `cos\ (x − 0.6435…)` | `= 1/2` | |
| `x − 0.6435…` | `= pi/3\ \ text(or)\ \ 2pi − pi/3` | |
| `x` | `= pi/3 + 0.6435…\ \ text(or)\ \ (5pi)/3 + 0.6435…` | |
| `= 1.691\ \ text(or)\ \ 5.879\ \ text{radians (to 3 d.p.)}` |
Express `5 cos x - 12 sin x` in the form `A cos (x + α)`, where `0 ≤ α ≤ pi/2`. (2 marks)
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`13\ cos\ (x + 1.176…)`
| `5\ cos\ x – 12\ sin\ x` | `= A\ cos\ (x + α)` |
| `= A\ cos\ x\ cos\ α – A\ sin\ x\ sin\ α` |
`:. A\ cos\ α = 5,\ \ \ A\ sin\ α = 12`
| `A^2` | `= 5^2 + 12^2 = 169` |
| `A` | `= 13` |
| `⇒ 13\ cos\ α` | `= 5` |
| `cos\ α` | `= 5/13` |
| `α` | `= cos^(-1)\ (5/13) ≈ 1.176…\ text(radians)` |
`:. 5cos x – 12sin x = 13 cos (x + 1.176…)`
Which expression is equal to `cos x - sin x`?
`A`
`R cos (x + alpha) = R cos x cos alpha – R sinx sin alpha`
`:.\ R cosx cos alpha – R sinx sin alpha = cos x – sin x`
`R cos alpha = 1,\ \ \ \ R sin alpha = 1`
| `R^2` | `= 1^2 + 1^2` |
| `R` | `= sqrt 2` |
| `:.\ sqrt 2 cos alpha` | `= 1` |
| `cos alpha` | `= 1/sqrt2` |
| `alpha` | `= pi/4` |
`:.\ sqrt 2 cos (x + pi/4) = cosx – sinx`
`=> A`
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i. `text(Write)\ sqrt3 cosx\ – sinx\ text(in form)`
`2cos (x + alpha),\ \ \ 0 < alpha < pi/2`
| `2 (cosx cos alpha\ – sinx sin alpha)` | `= sqrt 3 cosx\ – sinx` |
| `cosx cos alpha\ – sinx sin alpha` | `= sqrt3/2 cos x\ – 1/2 sinx` |
| `=> cos alpha` | `= sqrt3/2` |
| `=> sin alpha` | `= 1/2` |
| `alpha` | `= pi/6` |
`:.\ 2cos (x + pi/6) = sqrt3 cosx\ – sinx`
ii. `text(Solve)\ sqrt3 cosx = 1 + sinx,\ \ \ 0 < x < 2pi`
| `sqrt3 cosx\ – sinx` | `= 1` |
| `2 cos (x + pi/6)` | `= 1\ \ \ text{(from part (i))}` |
| `cos (x + pi/6)` | `= 1/2` |
| `cos(pi/3)` | `=1/2` |
`text(S)text(ince cos is positive in)\ 1^text(st) // 4^text(th)\ text(quadrants)`
| `x + pi/6` | `= pi/3,\ 2pi\ – pi/3\ \ \ \ \ (0 < x < 2pi)` |
| `:.\ x` | `= pi/6,\ (3pi)/2` |
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| i. | `2 cos theta + 2 cos (theta + pi/3)` |
| `= 2 cos theta + 2 (cos theta cos (pi/3)\ – sin theta sin (pi/3))` | |
| `= 2 cos theta + 2 cos theta xx 1/2\ – 2 sin theta xx sqrt3/2` | |
| `= 2 cos theta + cos theta\ – sqrt3 sin theta` | |
| `= 3 cos theta\ – sqrt3 sin theta` |
`R cos (theta + alpha) = R cos theta cos alpha – R sin theta sin alpha`
| `R cos alpha` | `= 3\ \ \ \ \ ` | `R sin alpha` | `= sqrt3` |
| `cos alpha` | `= 3/R\ \ \ \ \ ` | `sin alpha` | `= sqrt3/R` |
| `tan alpha` | `= sin alpha/cos alpha = sqrt3/3 = 1/sqrt3` |
| `tan\ pi/6` | `=1/sqrt3` |
| `:. alpha` | `=pi/6\ \ \ \ \ (0 < alpha < pi/2)` |
| `R^2` | `= 3^2 + (sqrt3)^2` |
| `= 9 + 3` | |
| `R` | `= sqrt 12 = 2 sqrt3` |
`:.\ 2 cos theta + 2 cos (theta + pi/3) = 2 sqrt 3 cos (theta + pi/6)`
| ii. | `2 cos theta + 2 cos(theta + pi/3)` | `= 3` |
| `2 sqrt 3 cos (theta + pi/6)` | `= 3` | |
| `cos (theta + pi/6)` | `= 3/(2sqrt3) = sqrt3/2` | |
| `cos^(-1) (sqrt3/2)` | `= pi/6` |
`text(S)text(ince cos is positive in 1st and 4th quadrants,)`
| `theta + pi/6` | `= pi/6,\ 2pi\ – pi/6` |
| `:. theta` | `= (5pi)/3\ \ \ \ \ (0 < theta < 2pi)` |