Understanding the Problem: sin 10° sin 50° sin 70°

We need to find the value of the trigonometric expression sin 10° sin 50° sin 70°. This involves multiplying the sine of three specific angles.

Trigonometric Identity for Sine Products

A useful trigonometric identity that simplifies products of sine functions is:

$$ \sin(60^\circ - \theta) \sin \theta \sin(60^\circ + \theta) = \dfrac{1}{4} \sin(3\theta) $$

This identity is particularly helpful when dealing with angles that are symmetrically spaced around 60 degrees.

Applying the Sine Product Identity

Let's compare the given expression sin 10° sin 50° sin 70° with the identity.

We can rewrite the expression as:

$$ \sin 10^\circ \sin 50^\circ \sin 70^\circ $$

Notice that the angles 10°, 50°, and 70° fit the pattern of the identity if we choose a specific value for θ (theta).

Let's set θ = 10°.

Then:

• θ is 10°
• 60° - θ becomes 60° - 10° = 50°
• 60° + θ becomes 60° + 10° = 70°

So, the expression sin 10° sin 50° sin 70° perfectly matches the left side of the identity with θ = 10°.

Step-by-Step Calculation

Using the identity, we can substitute θ = 10°:

$$ \sin 10^\circ \sin 50^\circ \sin 70^\circ = \sin(60^\circ - 10^\circ) \sin 10^\circ \sin(60^\circ + 10^\circ) $$

According to the identity, this is equal to:

$$ \dfrac{1}{4} \sin(3 \times 10^\circ) $$

Now, we simplify the argument of the sine function:

$$ 3 \times 10^\circ = 30^\circ $$

So the expression becomes:

$$ \dfrac{1}{4} \sin(30^\circ) $$

We know the standard value for sin(30°):

$$ \sin(30^\circ) = \dfrac{1}{2} $$

Substituting this value back:

$$ \dfrac{1}{4} \times \dfrac{1}{2} $$

Final Value

Performing the final multiplication:

$$ \dfrac{1}{4} \times \dfrac{1}{2} = \dfrac{1}{8} $$

Therefore, the value of sin 10° sin 50° sin 70° is 1/8.