The R-formula questions with answers

Questions

1. Question 1R and α

   Express $5\cos\theta - 12\sin\theta$ in the form $R\cos(\theta + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Find the exact value of $R$.

   Show the worked solution

   1. Step 1: Expand the R form with the addition formula, and compare coefficients$R\cos(\theta + \alpha) = R\cos\theta\cos\alpha - R\sin\theta\sin\alpha$$R\cos\alpha = 5$$R\sin\alpha = 12$
   2. Step 2: Square and add (cos²α + sin²α = 1) for R$R = \sqrt{5^{2} + 12^{2}}$$= 13$
2. Question 2R and α

   Express $2\cos\theta + 5\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Find the exact value of $R$.

   Show the worked solution

   1. Step 1: Expand the R form with the addition formula, and compare coefficients$R\cos(\theta - \alpha) = R\cos\theta\cos\alpha + R\sin\theta\sin\alpha$$R\cos\alpha = 2$$R\sin\alpha = 5$
   2. Step 2: Square and add (cos²α + sin²α = 1) for R$R = \sqrt{2^{2} + 5^{2}}$$= \sqrt{29}$
3. Question 3Maximum and minimum values

   Express $8\cos\theta + 15\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Hence find the value of $\theta$, where $0^\circ \leq \theta \leq 360^\circ$, at which $8\cos\theta + 15\sin\theta$ is greatest.

   Show the worked solution

   1. Step 1: Expand the R form with the addition formula, and compare coefficients$R\cos(\theta - \alpha) = R\cos\theta\cos\alpha + R\sin\theta\sin\alpha$$R\cos\alpha = 8$$R\sin\alpha = 15$
   2. Step 2: Square and add (cos²α + sin²α = 1) for R$R = \sqrt{8^{2} + 15^{2}}$$= 17$
   3. Step 3: Divide for tan α, then use tan⁻¹$\tan\alpha = \dfrac{15}{8}$$\alpha = 61.93^\circ$ (2 d.p.)
   4. Step 4: Write the expression in the R form, with the values of R and α$8\cos\theta + 15\sin\theta = 17\cos(\theta - 61.93^\circ)$
   5. Step 5: What is the greatest value that cos(θ − α) can take?The greatest value of $\cos(\theta - \alpha)$ is $1$.
   6. Step 6: Find the angle in the bracket that gives this value, then θ (with 0° ≤ θ ≤ 360°)$\theta - 61.93^\circ = 0^\circ$$\theta = 61.9^\circ$
4. Question 4Maximum and minimum values

   Express $2\sin\theta - 7\cos\theta$ in the form $R\sin(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Hence find the exact maximum value of $2\sin\theta - 7\cos\theta - 1$.

   Show the worked solution

   1. Step 1: Expand the R form with the addition formula, and compare coefficients$R\sin(\theta - \alpha) = R\sin\theta\cos\alpha - R\cos\theta\sin\alpha$$R\cos\alpha = 2$$R\sin\alpha = 7$
   2. Step 2: Square and add (cos²α + sin²α = 1) for R$R = \sqrt{2^{2} + 7^{2}}$$= \sqrt{53}$
   3. Step 3: Divide for tan α, then use tan⁻¹$\tan\alpha = \dfrac{7}{2}$$\alpha = 74.05^\circ$ (2 d.p.)
   4. Step 4: Write the expression in the R form, with the values of R and α$2\sin\theta - 7\cos\theta = \sqrt{53}\sin(\theta - 74.05^\circ)$
   5. Step 5: What is the greatest value that sin(θ − α) can take?The greatest value of $\sin(\theta - \alpha)$ is $1$.
   6. Step 6: So the maximum value of the expression isMaximum $= \sqrt{53} - 1$
5. Question 5Solving equations

   Express $3\sin\theta + 4\cos\theta$ in the form $R\sin(\theta + \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Hence solve $3\sin\theta + 4\cos\theta = -1$ for $0^\circ \leq \theta \leq 360^\circ$.

   Show the worked solution

   1. Step 1: Expand the R form with the addition formula, and compare coefficients$R\sin(\theta + \alpha) = R\sin\theta\cos\alpha + R\cos\theta\sin\alpha$$R\cos\alpha = 3$$R\sin\alpha = 4$
   2. Step 2: Square and add (cos²α + sin²α = 1) for R$R = \sqrt{3^{2} + 4^{2}}$$= 5$
   3. Step 3: Divide for tan α, then use tan⁻¹$\tan\alpha = \dfrac{4}{3}$$\alpha = 53.13^\circ$ (2 d.p.)
   4. Step 4: Write the expression in the R form, with the values of R and α$3\sin\theta + 4\cos\theta = 5\sin(\theta + 53.13^\circ)$
   5. Step 5: Divide both sides by R$\sin(\theta + 53.13^\circ) = -\dfrac{1}{5}$$= -0.2000\ldots$
   6. Step 6: Basic angle β: use the positive value$\beta = \sin^{-1}\left(0.2000\ldots\right)$$\beta = 11.5^\circ$
   7. Step 7: ASTC (All, Sine, Tan, Cos): in which quadrants does sin u have this sign?$\sin u$ is negative in quadrants 3 and 4.
   8. Step 8: List every angle u in the range (u = θ + α runs from α to 360° + α), quadrant by quadrant$u = 180^\circ + 11.5^\circ$$= 191.5^\circ$$u = 360^\circ - 11.5^\circ$$= 348.5^\circ$
   9. Step 9: Get θ back: θ = u − α$\theta = 138.4^\circ$$\theta = 295.3^\circ$
6. Question 6Solving equations

   Express $\cos\theta + 2\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0^\circ < \alpha < 90^\circ$. Hence solve $\cos\theta + 2\sin\theta = 2$ for $0^\circ \leq \theta \leq 360^\circ$.

   Show the worked solution

   1. Step 1: Expand the R form with the addition formula, and compare coefficients$R\cos(\theta - \alpha) = R\cos\theta\cos\alpha + R\sin\theta\sin\alpha$$R\cos\alpha = 1$$R\sin\alpha = 2$
   2. Step 2: Square and add (cos²α + sin²α = 1) for R$R = \sqrt{1^{2} + 2^{2}}$$= \sqrt{5}$
   3. Step 3: Divide for tan α, then use tan⁻¹$\tan\alpha = 2$$\alpha = 63.43^\circ$ (2 d.p.)
   4. Step 4: Write the expression in the R form, with the values of R and α$\cos\theta + 2\sin\theta = \sqrt{5}\cos(\theta - 63.43^\circ)$
   5. Step 5: Divide both sides by R$\cos(\theta - 63.43^\circ) = \dfrac{2}{\sqrt{5}}$$= 0.8944\ldots$
   6. Step 6: Basic angle β: use the positive value$\beta = \cos^{-1}\left(0.8944\ldots\right)$$\beta = 26.6^\circ$
   7. Step 7: ASTC (All, Sine, Tan, Cos): in which quadrants does cos u have this sign?$\cos u$ is positive in quadrants 1 and 4.
   8. Step 8: List every angle u in the range (u = θ − α runs from −α to 360° − α), quadrant by quadrant$u = -26.6^\circ$$= -26.6^\circ$$u = 26.6^\circ$
   9. Step 9: Get θ back: θ = u + α$\theta = 36.9^\circ$$\theta = 90.0^\circ$