Answer :
Sure! Let's match the trigonometric expressions to their solutions step-by-step.
1. First expression: [tex]$\cos \left[140^{\circ}-\left(50^{\circ}+30^{\circ}\right)\right]$[/tex]
We need to simplify the angle inside the cosine function.
[tex]\[ 140^{\circ} - (50^{\circ} + 30^{\circ}) = 140^{\circ} - 80^{\circ} = 60^{\circ} \][/tex]
So, the expression simplifies to [tex]$\cos(60^{\circ})$[/tex].
The value of [tex]$\cos(60^{\circ})$[/tex] is [tex]$\frac{1}{2}$[/tex].
2. Second expression: [tex]$\tan 240^{\circ}$[/tex]
The value of [tex]$\tan(240^{\circ})$[/tex] is [tex]$\sqrt{3}$[/tex].
3. Third expression: [tex]$\sin 255^{\circ}$[/tex]
The value of [tex]$\sin(255^{\circ})$[/tex] is [tex]$\frac{-\sqrt{6}-\sqrt{2}}{4}$[/tex].
4. Fourth expression: [tex]$\cos 150^{\circ}$[/tex]
The value of [tex]$\cos(150^{\circ})$[/tex] is [tex]$-\frac{\sqrt{3}}{2}$[/tex].
To summarize, the pairs should be matched as follows:
1. [tex]$\cos \left[140^{\circ}-\left(50^{\circ}+30^{\circ}\right)\right]$[/tex] matches with [tex]$\frac{1}{2}$[/tex]
2. [tex]$\sqrt{3}$[/tex] matches with [tex]$\tan 240^{\circ}$[/tex]
3. [tex]$\sin 255^{\circ}$[/tex] matches with [tex]$\frac{-\sqrt{6}-\sqrt{2}}{4}$[/tex]
4. [tex]$\cos 150^{\circ}$[/tex] matches with [tex]$-\frac{\sqrt{3}}{2}$[/tex]
So, the final pairs are:
1. [tex]$\cos \left[140^{\circ}-\left(50^{\circ}+30^{\circ}\right)\right]$[/tex] [tex]$\to \frac{1}{2}$[/tex]
2. [tex]$\tan 240^{\circ}$[/tex] [tex]$\to \sqrt{3}$[/tex]
3. [tex]$\sin 255^{\circ}$[/tex] [tex]$\to \frac{-\sqrt{6}-\sqrt{2}}{4}$[/tex]
4. [tex]$\cos 150^{\circ}$[/tex] [tex]$\to -\frac{\sqrt{3}}{2}$[/tex]
1. First expression: [tex]$\cos \left[140^{\circ}-\left(50^{\circ}+30^{\circ}\right)\right]$[/tex]
We need to simplify the angle inside the cosine function.
[tex]\[ 140^{\circ} - (50^{\circ} + 30^{\circ}) = 140^{\circ} - 80^{\circ} = 60^{\circ} \][/tex]
So, the expression simplifies to [tex]$\cos(60^{\circ})$[/tex].
The value of [tex]$\cos(60^{\circ})$[/tex] is [tex]$\frac{1}{2}$[/tex].
2. Second expression: [tex]$\tan 240^{\circ}$[/tex]
The value of [tex]$\tan(240^{\circ})$[/tex] is [tex]$\sqrt{3}$[/tex].
3. Third expression: [tex]$\sin 255^{\circ}$[/tex]
The value of [tex]$\sin(255^{\circ})$[/tex] is [tex]$\frac{-\sqrt{6}-\sqrt{2}}{4}$[/tex].
4. Fourth expression: [tex]$\cos 150^{\circ}$[/tex]
The value of [tex]$\cos(150^{\circ})$[/tex] is [tex]$-\frac{\sqrt{3}}{2}$[/tex].
To summarize, the pairs should be matched as follows:
1. [tex]$\cos \left[140^{\circ}-\left(50^{\circ}+30^{\circ}\right)\right]$[/tex] matches with [tex]$\frac{1}{2}$[/tex]
2. [tex]$\sqrt{3}$[/tex] matches with [tex]$\tan 240^{\circ}$[/tex]
3. [tex]$\sin 255^{\circ}$[/tex] matches with [tex]$\frac{-\sqrt{6}-\sqrt{2}}{4}$[/tex]
4. [tex]$\cos 150^{\circ}$[/tex] matches with [tex]$-\frac{\sqrt{3}}{2}$[/tex]
So, the final pairs are:
1. [tex]$\cos \left[140^{\circ}-\left(50^{\circ}+30^{\circ}\right)\right]$[/tex] [tex]$\to \frac{1}{2}$[/tex]
2. [tex]$\tan 240^{\circ}$[/tex] [tex]$\to \sqrt{3}$[/tex]
3. [tex]$\sin 255^{\circ}$[/tex] [tex]$\to \frac{-\sqrt{6}-\sqrt{2}}{4}$[/tex]
4. [tex]$\cos 150^{\circ}$[/tex] [tex]$\to -\frac{\sqrt{3}}{2}$[/tex]
