The exact value of sin 15° is [tex]\frac{\sqrt{3}-1}{2\sqrt{2}}[/tex].
Solution:
45 – 30 = 15
sin 15° can be written as sin (45° – 30°).
Using trigonometric difference formula:
sin(A – B) = sin A cos B – cos A sin B
sin (45° – 30°) = sin 45° cos 30° – cos 45° sin 30°
The value of sin 45° = [tex]\frac{1}{\sqrt{2}}[/tex]
The value of cos 30° = [tex]\frac{\sqrt{3}}{2}[/tex]
The value of cos 45° = [tex]\frac{1}{\sqrt{2}}[/tex]
The value of sin 30° = [tex]\frac{1}{2}[/tex]
Substitute these values, we get
[tex]$\sin \left(45^{\circ}-30^{\circ}\right)=\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \cdot \frac{1}{2}[/tex]
[tex]$=\frac{\sqrt{3}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}[/tex]
[tex]$=\frac{\sqrt{3}-1}{2\sqrt{2}}[/tex]
[tex]$\sin \left(45^{\circ}-30^{\circ}\right)=\frac{\sqrt{3}-1}{2\sqrt{2}}[/tex]
[tex]$\sin15^\circ}=\frac{\sqrt{3}-1}{2\sqrt{2}}[/tex]
Hence the exact value of sin 15° is [tex]\frac{\sqrt{3}-1}{2\sqrt{2}}[/tex].
