Question Statement

Find the general solution, in radians, of the following equations:

(i) $10\cos x-24\sin x=13$ (ii) $8\sin x+15\cos x=6$ (iii) $\cos x-\sin 4x=0$ (iv) $5\sin x+3\cos x=4$ (v) $4\cos x+3\sin x=2$ (vi) $\sin 7x-\sin x=\cos 4x$ (vii) $\tan k=3\sin 2k$ (viii) $5\sin 3\beta +6\cos 3\beta =1$ (ix) $2\cos 4x+8\cos 2x=3$ (x) $\tan 3x=\sqrt{3}$ (xi) ${\tan }^{2}4x=3$ (xii) $\sec \mu +\tan \mu =3$ (xiii) $4\sin 2x-3\cos 2x=2$ (xiv) $\cos 4x+\cos 6x=\cos 5x$ (xv) $4{\cos }^{2}x+1=4\cos x$ (xvi) $3{\tan }^{2}x+5\sec x+1=0$ (xvii) $\sqrt{2}\tan x\cos x-\tan x=0$

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Background and Explanation

To solve trigonometric equations of the form $a\sin x+b\cos x=c$, we often use the method of squaring both sides to create a quadratic equation in terms of a single trigonometric function (using ${\sin }^{2}x+{\cos }^{2}x=1$). However, squaring can introduce extraneous solutions, so all results must be checked against the original equation. For equations involving multiple angles, we use sum-to-product identities or double-angle formulas to factorize the expression.

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Solution

Part (i)

$10\cos x-24\sin x=13$

Isolate one term and square: $10\cos x=13+24\sin x$ $(10\cos x{)}^2=(13+24\sin x{)}^2$ $100{\cos }^{2}x=169+576{\sin }^{2}x+624\sin x$ $100(1-{\sin }^{2}x)=169+576{\sin }^{2}x+624\sin x$ $676{\sin }^{2}x+624\sin x+69=0$

Using the quadratic formula: $\sin x=\frac{-624\pm \sqrt{(624{)}^2-4(676)(69)}}{2\times 676}=\frac{-624\pm \sqrt{202800}}{1352}$ $\sin x\approx \frac{-624\pm 450.33}{1352}$

Case 1: $\sin x\approx -0.13$ Reference angle $\approx 7.38^{\circ }=\frac{7.38\pi }{180}$ rad.

• 3rd Quad: $x=\pi +\frac{7.38\pi }{180}$ (Check: Not satisfied)
• 4th Quad: $x=2\pi -\frac{7.38\pi }{180}=\frac{352.62\pi }{180}$ (Check: Satisfied)

Case 2: $\sin x\approx -0.79$ Reference angle $\approx 52.18^{\circ }=\frac{52.18\pi }{180}$ rad.

• 3rd Quad: $x=\pi +\frac{52.18\pi }{180}=\frac{232.18\pi }{180}$ (Check: Satisfied)
• 4th Quad: $x=2\pi -\frac{52.18\pi }{180}$ (Check: Not satisfied)

General Solution: $x=\frac{232.18\pi }{180}+2n\pi$ and $x=\frac{352.62\pi }{180}+2n\pi$

Part (ii)

$8\sin x+15\cos x=6$

$15\cos x=6-8\sin x$ $225(1-{\sin }^{2}x)=36+64{\sin }^{2}x-96\sin x$ $289{\sin }^{2}x-96\sin x-189=0$ $\sin x=\frac{48\pm 15\sqrt{253}}{289}$

Case 1: $\sin x\approx 0.9916$

• 1st Quad: $x=\frac{82.56\pi }{180}$ (Check: Not satisfied)
• 2nd Quad: $x=\pi -\frac{82.56\pi }{180}=\frac{97.44\pi }{180}$ (Check: Satisfied)

Case 2: $\sin x\approx -0.6595$

• 3rd Quad: $x=\pi +\frac{41.26\pi }{180}$ (Check: Not satisfied)
• 4th Quad: $x=2\pi -\frac{41.26\pi }{180}=\frac{318.74\pi }{180}$ (Check: Satisfied)

General Solution: $x=\frac{97.44\pi }{180}+2n\pi \cup \frac{318.74\pi }{180}+2n\pi$

Part (iii)

$\cos x-\sin 4x=0$

$\cos x-2\sin 2x\cos 2x=0$ $\cos x-2(2\sin x\cos x)\cos 2x=0$ $\cos x(1-4\sin x\cos 2x)=0$

Case 1: $\cos x=0$ $x=\frac{\pi }{2}+2n\pi ,\frac{3\pi }{2}+2n\pi$

Case 2: $1-4\sin x(1-2{\sin }^{2}x)=0\Rightarrow 8{\sin }^{3}x-4\sin x+1=0$ By inspection, $\sin x=1/2$ is a root. Using synthetic division:

Depressed equation: $4{\sin }^{2}x+2\sin x-1=0$

• From $\sin x=1/2$: $x=\frac{\pi }{6}+2n\pi ,\frac{5\pi }{6}+2n\pi$
• From $4{\sin }^{2}x+2\sin x-1=0$: $\sin x=\frac{-1\pm \sqrt{5}}{4}$
  • $\sin x\approx 0.309\Rightarrow x=\frac{\pi }{10}+2n\pi ,\frac{9\pi }{10}+2n\pi$
  • $\sin x\approx -0.809\Rightarrow x=\frac{13\pi }{10}+2n\pi ,\frac{17\pi }{10}+2n\pi$

Part (iv)

$5\sin x+3\cos x=4$

Squaring leads to $34{\sin }^{2}x-40\sin x+7=0$. $\sin x=\frac{20\pm 9\sqrt{2}}{34}$

• $\sin x\approx 0.9625\Rightarrow x=\frac{105.74\pi }{180}$ (Satisfied)
• $\sin x\approx 0.2138\Rightarrow x=\frac{12.34\pi }{180}$ (Satisfied)

General Solution: $x=\frac{105.74\pi }{180}+2n\pi \cup \frac{12.34\pi }{180}+2n\pi$

Part (v)

$4\cos x+3\sin x=2$

Squaring leads to $25{\sin }^{2}x-12\sin x-12=0$. $\sin x=\frac{6\pm 4\sqrt{21}}{25}$

• $\sin x\approx 0.9732\Rightarrow x=\frac{103.30\pi }{180}$ (Satisfied)
• $\sin x\approx -0.4932\Rightarrow x=\frac{330.45\pi }{180}$ (Satisfied)

General Solution: $x=\frac{103.30\pi }{180}+2n\pi \cup \frac{330.45\pi }{180}+2n\pi$

Part (vi)

$\sin 7x-\sin x=\cos 4x$

Using $\sin \alpha -\sin \beta =2\cos (\frac{\alpha +\beta }{2})\sin (\frac{\alpha -\beta }{2})$: $2\cos 4x\sin 3x=\cos 4x$ $\cos 4x(2\sin 3x-1)=0$

• $\cos 4x=0\Rightarrow 4x=\frac{\pi }{2},\frac{3\pi }{2}\Rightarrow x=\frac{\pi }{8}+\frac{n\pi }{2}$
• $\sin 3x=1/2\Rightarrow 3x=\frac{\pi }{6},\frac{5\pi }{6}\Rightarrow x=\frac{\pi }{18}+\frac{2n\pi }{3},\frac{5\pi }{18}+\frac{2n\pi }{3}$

Part (vii)

$\tan k=3\sin 2k$

$\frac{\sin k}{\cos k}=6\sin k\cos k$ $\sin k(1-6{\cos }^{2}k)=0$

• $\sin k=0\Rightarrow k=n\pi$
• ${\cos }^{2}k=1/6\Rightarrow \cos k=\pm \frac{1}{\sqrt{6}}$
  • $k\approx \frac{65.90\pi }{180},\frac{114.10\pi }{180},\frac{245.90\pi }{180},\frac{294.10\pi }{180}$ (all $+2n\pi$)

Part (viii)

$5\sin 3\beta +6\cos 3\beta =1$

Squaring leads to $61{\sin }^{2}3\beta -10\sin 3\beta -35=0$. $\sin 3\beta =\frac{5\pm 12\sqrt{15}}{61}$

• $\sin 3\beta \approx 0.8438\Rightarrow 3\beta =\frac{122.46\pi }{180}\Rightarrow \beta =\frac{40.82\pi }{180}$
• $\sin 3\beta \approx -0.6799\Rightarrow 3\beta =\frac{317.16\pi }{180}\Rightarrow \beta =\frac{105.72\pi }{180}$

General Solution: $\beta =\frac{40.82\pi }{180}+2n\pi \cup \frac{105.72\pi }{180}+2n\pi$

Part (ix)

$2\cos 4x+8\cos 2x=3$

$2(2{\cos }^{2}2x-1)+8\cos 2x-3=0$ $4{\cos }^{2}2x+8\cos 2x-5=0$ Let $u=\cos 2x$: $4u^2+8u-5=0\Rightarrow (2u-1)(2u+5)=0$

• $\cos 2x=1/2\Rightarrow 2x=\frac{\pi }{3},\frac{5\pi }{3}\Rightarrow x=\frac{\pi }{6}+n\pi ,\frac{5\pi }{6}+n\pi$
• $\cos 2x=-5/2$ (Impossible)

Part (x)

$\tan 3x=\sqrt{3}$

$3x=\frac{\pi }{3}+n\pi \Rightarrow x=\frac{\pi }{9}+\frac{n\pi }{3}$

Part (xi)

${\tan }^{2}4x=3\Rightarrow \tan 4x=\pm \sqrt{3}$

• $4x=\frac{\pi }{3}+n\pi \Rightarrow x=\frac{\pi }{12}+\frac{n\pi }{4}$
• $4x=-\frac{\pi }{3}+n\pi \Rightarrow x=-\frac{\pi }{12}+\frac{n\pi }{4}$

Part (xii)

$\sec \mu +\tan \mu =3$

$\frac{1+\sin \mu }{\cos \mu }=3\Rightarrow (1+\sin \mu {)}^2=9(1-{\sin }^2\mu )$ $10{\sin }^2\mu +2\sin \mu -8=0\Rightarrow (\sin \mu +1)(5\sin \mu -4)=0$

• $\sin \mu =-1\Rightarrow \mu =\frac{3\pi }{2}$ (Extraneous, $\sec$ undefined)
• $\sin \mu =0.8\Rightarrow \mu =\frac{53.13\pi }{180}+2n\pi$ (Satisfied)

Part (xiii)

$4\sin 2x-3\cos 2x=2$

Squaring leads to $25{\sin }^{2}2x-16\sin 2x-5=0$. $\sin 2x=\frac{16\pm 6\sqrt{21}}{50}$

• $\sin 2x\approx 0.8699\Rightarrow 2x=60.44^{\circ }\Rightarrow x=\frac{30.22\pi }{180}$
• $\sin 2x\approx -0.2299\Rightarrow 2x=193.29^{\circ }\Rightarrow x=\frac{96.64\pi }{180}$

Part (xiv)

$\cos 4x+\cos 6x=\cos 5x$

$2\cos 5x\cos x=\cos 5x\Rightarrow \cos 5x(2\cos x-1)=0$

• $\cos 5x=0\Rightarrow 5x=(2n+1)\frac{\pi }{2}\Rightarrow x=(2n+1)\frac{\pi }{10}$
• $\cos x=1/2\Rightarrow x=\frac{\pi }{3}+2n\pi ,\frac{5\pi }{3}+2n\pi$

Part (xv)

$4{\cos }^{2}x-4\cos x+1=0$

$(2\cos x-1{)}^2=0\Rightarrow \cos x=1/2$ $x=\frac{\pi }{3}+2n\pi ,\frac{5\pi }{3}+2n\pi$

Part (xvi)

$3{\tan }^{2}x+5\sec x+1=0$

$3({\sec }^{2}x-1)+5\sec x+1=0\Rightarrow 3{\sec }^{2}x+5\sec x-2=0$ $(3\sec x-1)(\sec x+2)=0$

• $\sec x=1/3\Rightarrow \cos x=3$ (Impossible)
• $\sec x=-2\Rightarrow \cos x=-1/2\Rightarrow x=\frac{2\pi }{3}+2n\pi ,\frac{4\pi }{3}+2n\pi$

Part (xvii)

$\sqrt{2}\tan x\cos x-\tan x=0$

$\tan x(\sqrt{2}\cos x-1)=0$

• $\tan x=0\Rightarrow x=n\pi$
• $\cos x=\frac{1}{\sqrt{2}}\Rightarrow x=\frac{\pi }{4}+2n\pi ,\frac{7\pi }{4}+2n\pi$

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Key Formulas or Methods Used

• Pythagorean Identity: ${\sin }^{2}x+{\cos }^{2}x=1$ or ${\tan }^{2}x+1={\sec }^{2}x$
• Double Angle Formulas:
  • $\sin 2x=2\sin x\cos x$
  • $\cos 2x=2{\cos }^{2}x-1=1-2{\sin }^{2}x$
• Sum-to-Product Identities:
  • $\sin \alpha -\sin \beta =2\cos (\frac{\alpha +\beta }{2})\sin (\frac{\alpha -\beta }{2})$
  • $\cos \alpha +\cos \beta =2\cos (\frac{\alpha +\beta }{2})\cos (\frac{\alpha -\beta }{2})$
• Quadratic Formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
• General Solution: Adding $2n\pi$ for $\sin /\cos$ and $n\pi$ for $\tan$.

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Summary of Steps

1. Simplify the Equation: Use identities to reduce the equation to a single trigonometric function or a product of factors.
2. Solve for the Trig Function: Use algebraic methods like factoring or the quadratic formula.
3. Find Reference Angles: Determine the basic angle in the first quadrant.
4. Identify Quadrants: Based on the sign (positive/negative), find the angles in the relevant quadrants ($[0,2\pi ]$).
5. Check for Extraneous Solutions: Especially important if you squared both sides during step 1.
6. Write General Solution: Add the appropriate period ($2n\pi$ or $n\pi$) to the valid solutions.