Exercicis de trigonometria V
Exercici 1
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle 5\sin x + 3 = 3\sin x + 4 \)
Solució:
\(\displaystyle
\begin{align}
5\sin x + 3 = 3\sin x + 4
\quad & \Rightarrow \quad 2\sin x = 1 \\[10pt]
\quad & \Rightarrow \quad \sin x = \frac{1}{2} \\[10pt]
\quad & \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=30^{\circ}\\x=150^{\circ}
\end{array}\right.
\end{align}
\)
Exercici 2
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle 3\sin^2 x - \cos^2 x = 2 \)
Solució:
\(\displaystyle
\begin{align}
3\sin^2 x-\cos^2 x=2
\quad & \Rightarrow \quad 3\sin^2 x-\left(1-\sin^2 x\right)=2 \\[10pt]
\quad & \Rightarrow \quad 3\sin^2 x-1+\sin^2 x=2 \\[10pt]
\quad & \Rightarrow \quad 4\sin^2 x=3 \\[10pt]
\quad & \Rightarrow \quad \sin^2 x=\frac{3}{4} \\[10pt]
\quad & \Rightarrow \quad \sin x= \pm\frac{\sqrt{3}}{2}
\end{align}
\)
\(\displaystyle
\sin x= +\frac{\sqrt{3}}{2}
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=60^{\circ}\\x=120^{\circ}
\end{array}\right.
\)
\(\displaystyle
\sin x= -\frac{\sqrt{3}}{2}
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=240^{\circ}\\x=300^{\circ}
\end{array}\right.
\)
Exercici 3
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle \tan\left(90^{\circ}-x\right)=\frac{\tan x}{3}\)
Solució:
\(\displaystyle
\begin{align}
\tan\left(90^{\circ}-x\right)=\frac{\tan x}{3}
\quad & \Rightarrow \quad \frac{1}{\tan x}=\frac{\tan x}{3} \\[10pt]
\quad & \Rightarrow \quad \tan^2 x=3 \\[10pt]
\quad & \Rightarrow \quad \tan x=\pm\sqrt{3}
\end{align}
\)
\(\displaystyle
\tan x=+\sqrt{3}
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=60^{\circ}\\x=240^{\circ}
\end{array}\right.
\)
\(\displaystyle
\tan x=-\sqrt{3}
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=120^{\circ}\\x=300^{\circ}
\end{array}\right.
\)
Exercici 4
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle \cos^2 x = \cos x \)
Solució:
\(\displaystyle
\begin{align}
\cos^2 x=\cos x
\quad & \Rightarrow \quad \cos^2 x-\cos x=0 \\[10pt]
\quad & \Rightarrow \quad \cos x \cdot \left( \cos x - 1 \right) = 0 \\[10pt]
\quad & \Rightarrow \quad
\left\lbrace\begin{array}{l}
\cos x=0\\\cos x=1
\end{array}\right.
\end{align}
\)
\(\displaystyle
\cos x=0
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=90^{\circ}\\x=270^{\circ}
\end{array}\right.
\)
\(\displaystyle
\cos x=1
\quad \Rightarrow \quad
x=0^{\circ}
\)
Exercici 5
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle 2\cos^2 x+3\cos x+1=0\)
Solució:
\(\displaystyle
\cos x =
\frac{-3\pm\sqrt{9-8}}{4} =
\frac{-3\pm1}{4} =
\left\lbrace\begin{array}{l}
\cos x=-\frac{1}{2} \\[5pt] \cos x=-1
\end{array}\right.
\)
\(\displaystyle
\cos x=-\frac{1}{2}
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=120^{\circ}\\x=240^{\circ}
\end{array}\right.
\)
\(\displaystyle
\cos x=-1
\quad \Rightarrow \quad
x=180^{\circ}
\)
Exercici 6
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle 5\sin x \cos x= \sin x \)
Solució:
\(\displaystyle
\begin{align}
5 \sin x \cos x= \sin x
\quad & \Rightarrow \quad 5\sin x \cos x - \sin x = 0 \\[10pt]
\quad & \Rightarrow \quad \sin x \cdot \left( 5\cos x - 1 \right) = 0 \\[10pt]
\quad & \Rightarrow \quad
\left\lbrace\begin{array}{l}
\sin x=0 \\[5pt] \cos x=\frac{1}{5}
\end{array}\right.
\end{align}
\)
\(\displaystyle
\sin x=0
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=0^{\circ}\\x=180^{\circ}
\end{array}\right.
\)
\(\displaystyle
\cos x=\frac{1}{5}
\quad \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=\text{78,46}^{\circ}\\x=\text{281,54}^{\circ}
\end{array}\right.
\)
Exercici 7
Resol la següent equació trigonomètrica, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
\(\displaystyle \cos^2 \left( \frac{x}{2} \right) + \cos x = 1 \)
Solució:
\(\displaystyle
\begin{align}
\cos^2 \left( \frac{x}{2} \right) + \cos x = 1
\quad & \Rightarrow \quad \left( \sqrt{\frac{1+\cos x}{2}} \right)^2 + \cos x = 1 \\[10pt]
\quad & \Rightarrow \quad \frac{1+\cos x}{\cancel{2}} + \frac{2\cos x}{\cancel{2}} = \frac{2}{\cancel{2}} \\[10pt]
\quad & \Rightarrow \quad 3\cos x = 1 \\[10pt]
\quad & \Rightarrow \quad \cos x = \frac{1}{3} \\[10pt]
\quad & \Rightarrow \quad
\left\lbrace\begin{array}{l}
x=\text{70,53}^{\circ}\\x=\text{289,47}^{\circ}
\end{array}\right.
\end{align}
\)
Exercici 8
Resol les següents equacions trigonomètriques, donant totes les solucions compreses entre \(0^{\circ}\) i \(360^{\circ}\).
| a) \(\displaystyle 2\sin^2 x + 3\sin x = 2 \) |
Solució: |
\(x_1 = 30^{\circ}\) i \(x_2 = 150^{\circ}\)
|
| b) \(\displaystyle 3\tan^2 x - 4\sqrt{3}\tan x + 3 = 0 \) |
Solució: |
\(x_1 = 30^{\circ}\), \(x_2 = 60^{\circ}\), \(x_3 = 210^{\circ}\) i \(x_4 = 240^{\circ}\)
|
| c) \(\displaystyle 3\sin x - \frac{1}{\sin x} = 2 \) |
Solució: |
\(x_1 = 90^{\circ}\), \(x_2 = \text{199,47}^{\circ}\) i \(x_3 = \text{340,53}^{\circ}\)
|
| d) \(\displaystyle 2\cos^2 x - \sqrt{3}\cos x = 0 \) |
Solució: |
\(x_1 = 30^{\circ}\), \(x_2 = 90^{\circ}\), \(x_3 = 270^{\circ}\) i \(x_4 = 330^{\circ}\)
|
| e) \(\displaystyle 3\left( 1-\cos x \right) = \sin^2 x\) |
Solució: |
\(x_1 = 0^{\circ}\)
|
Exercici 9
Troba tots els valors \( x \in \left[ 0^{\circ}, 360^{\circ} \right) \) que verifiquen la següent equació trigonomètrica:
\(\displaystyle \cos (2x)-\cos x = 0 \)
Solució:
\(\displaystyle
\begin {align}
\cos (2x)-\cos x = 0
&\quad\Rightarrow\quad \cos^2 x - \sin^2 x - \cos x = 0 \\[8pt]
&\quad\Rightarrow\quad \cos^2 x - \sin^2 x - \cos x = 0 \\[8pt]
&\quad\Rightarrow\quad \cos^2 x - (1-\cos^2 x) - \cos x = 0 \\[8pt]
&\quad\Rightarrow\quad 2\cos^2 x - \cos x - 1 = 0 \\[8pt]
&\quad\Rightarrow\quad \cos x = \dfrac{1\pm\sqrt{1+8}}{4} = \dfrac{1\pm3}{4}
\Rightarrow \left\lbrace\begin{array}{lll}
\cos x=1 &\quad\Rightarrow\quad x=0^{\circ} \\[6pt]
\cos x=-\frac{1}{2} &\quad\Rightarrow \left\lbrace\begin{array}{l} x=120^{\circ}\\[6pt] x=240^{\circ} \end{array}\right.
\end{array}\right.\\[8pt]
\end {align}
\)
Exercici 10
Troba tots els valors \( x \in \left[ 0^{\circ}, 360^{\circ} \right) \) que verifiquen la següent equació trigonomètrica:
\(\displaystyle \sin (2x)-\cos x = 0 \)
Solució:
\(\displaystyle
\begin {align}
\sin (2x)-\cos x = 0
&\quad\Rightarrow\quad 2 \sin x \cos x - \cos x = 0 \\[8pt]
&\quad\Rightarrow\quad \cos x \cdot (2\sin x - 1) = 0 \quad \Rightarrow \quad \left\lbrace\begin{array}{lll}
\cos x=0 &\quad\Rightarrow\quad \left\lbrace\begin{array}{l} x=90^{\circ}\\[6pt] x=270^{\circ} \end{array}\right. \\[10pt]
\sin x=\frac{1}{2} &\quad\Rightarrow \left\lbrace\begin{array}{l} x=30^{\circ}\\[6pt] x=150^{\circ} \end{array}\right.
\end{array}\right.\\[8pt]
\end {align}
\)