What is the value of sin75°?

The value of sin75° is [Tex]\frac{\sqrt{6} – \sqrt{2}}{4}[/Tex]​​ or approximately 0.9659.

To find the value of [Tex]\bold{\sin 75^\circ}[/Tex], we can express [Tex]\bold{75^\circ}[/Tex] as the sum of [Tex]\bold{45^\circ}[/Tex] and [Tex]\bold{30^\circ}[/Tex]. Using the sum-to-product trigonometric identity, which states that

sin(A + B) = sin A cos B + cos A sin B

we get:

[Tex]\bold{\sin 75^\circ = \sin (45^\circ + 30^\circ)}[/Tex]

Now, applying the identity:

[Tex]\bold{\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ}[/Tex]

Here, we use the known values: [Tex]\bold{\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}}[/Tex] and [Tex]\bold{\cos 30^\circ = \frac{\sqrt{3}}{2}}[/Tex]​​ and [Tex]\bold{\sin 30^\circ = \frac{1}{2}}[/Tex].

[Tex]\bold{\sin 75^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}[/Tex]

[Tex]\bold{\sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}}[/Tex]

This is the exact value of [Tex]\bold{\sin 75^\circ}[/Tex]. In decimal approximation, it is approximately 0.9659. This process involves breaking down the angle into known values, applying trigonometric identities, and simplifying to obtain the final result.