Determine the value of the actute angle $$ x $$ in each of the following equations. $$ \begin{array}{ll} ext { (1) } \sin \left(x-20^{\circ} ight)=0,678 & ext { (2) } 3 \cos \left(x+30^{\circ} ight)=2,121 \ ext { (3) } 2 an \left(2 x-10^{\circ} ight)=3,4641 & ext { (4) } 2 an 2 x-10=3,4641\end{array} $$

Question

Determine the value of the actute angle $$ x $$ in each of the following equations. $$ \begin{array}{ll}	ext { (1) } \sin \left(x-20^{\circ}ight)=0,678 & 	ext { (2) } 3 \cos \left(x+30^{\circ}ight)=2,121 \ 	ext { (3) } 2 	an \left(2 x-10^{\circ}ight)=3,4641 & 	ext { (4) } 2 	an 2 x-10=3,4641\end{array} $$

UpStudy AI · Accepted Answer

**(1)** We are given  
$$
\sin\left(x-20^\circ\right)=0.678.
$$  
Let  
$$
z = x-20^\circ,
$$  
so that  
$$
\sin z = 0.678.
$$  
Taking the inverse sine we have  
$$
z = \sin^{-1}(0.678) \approx 42.5^\circ.
$$  
Thus,  
$$
x = z + 20^\circ \approx 42.5^\circ + 20^\circ = 62.5^\circ.
$$  
(We discard the second solution $$180^\circ-42.5^\circ=137.5^\circ$$ because then $$x=157.5^\circ$$ is not acute.)

---

**(2)** We are given  
$$
3\cos\left(x+30^\circ\right)=2.121.
$$  
Divide both sides by 3:  
$$
\cos\left(x+30^\circ\right)=\frac{2.121}{3}\approx 0.707.
$$  
Since  
$$
\cos 45^\circ \approx 0.707,
$$  
we have  
$$
x+30^\circ=45^\circ \quad \Longrightarrow \quad x=45^\circ-30^\circ=15^\circ.
$$

---

**(3)** We are given  
$$
2\tan\left(2x-10^\circ\right)=3.4641.
$$  
Divide by 2:  
$$
\tan\left(2x-10^\circ\right)=\frac{3.4641}{2}\approx1.73205.
$$  
Since  
$$
\tan60^\circ = \sqrt{3}\approx1.73205,
$$  
we write  
$$
2x-10^\circ=60^\circ \quad \Longrightarrow \quad 2x=70^\circ \quad \Longrightarrow \quad x=35^\circ.
$$

---

**(4)** We are given  
$$
2\tan 2x-10=3.4641.
$$  
This is interpreted as  
$$
2\tan2x - 10 = 3.4641.
$$
Add 10 to both sides:  
$$
2\tan2x = 3.4641 + 10 = 13.4641.
$$  
Then divide by 2:  
$$
\tan2x = \frac{13.4641}{2} \approx6.73205.
$$
Now, take the inverse tangent:  
$$
2x = \tan^{-1}(6.73205).
$$
A quick estimation shows that  
$$
\tan81.5^\circ\approx6.75,
$$
so we approximate  
$$
2x\approx81.5^\circ,
$$  
which gives  
$$
x\approx\frac{81.5^\circ}{2}\approx40.75^\circ.
$$
Rounding to one decimal place, we can write  
$$
x\approx40.7^\circ.
$$

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**Summary of Answers:**

(1) $$x\approx62.5^\circ$$  
(2) $$x\approx15^\circ$$  
(3) $$x\approx35^\circ$$  
(4) $$x\approx40.7^\circ$$
The acute angles are: 1. $$ x \approx 62.5^\circ $$ 2. $$ x \approx 15^\circ $$ 3. $$ x \approx 35^\circ $$ 4. $$ x \approx 40.7^\circ $$

Determine the value of the actute angle in each of the following equations.

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Equation (1)

Equation (2)

Equation (3)

Equation (4)

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