Question

Value of \(\sin \frac{\pi}{6} \sin \frac{2 \pi}{6} \sin \frac{4 \pi}{6} \sin \frac{5 \pi}{6}\) is (a) \(\frac{3}{8}\) (b) \(\frac{3}{16}\) (c) \(\frac{\sqrt{3}}{8}\) (d) \(\frac{3}{4}\)

Short answer

Question: Find the value of \(\sin \frac{\pi}{6} \sin \frac{2 \pi}{6} \sin \frac{4 \pi}{6} \sin \frac{5 \pi}{6}\). Answer: \(\frac{3}{16}\)

Step-by-step solution

Simplify the angles given

First, let's simplify the angles: $\sin \frac{\pi}{6} = \sin \frac{1 \pi}{6} \\ \sin \frac{2 \pi}{6} = \sin \frac{1 \pi}{3}\\ \sin \frac{4 \pi}{6} = \sin \frac{2 \pi}{3}\\ \sin \frac{5 \pi}{6}$ Now, we have \(\sin \frac{1 \pi}{6} \sin \frac{1 \pi}{3} \sin \frac{2 \pi}{3} \sin \frac{5 \pi}{6}\).

Use known Trigonometric Identity values

We know the values of \(\sin\) at the angles \(\frac{\pi}{6}\) and \(\frac{\pi}{3}\) from the Unit Circle: \(\sin \frac{\pi}{6} = \frac{1}{2} \quad\) and \(\quad \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\) Now, we need to find values for \(\sin \frac{2 \pi}{3}\) and \(\sin \frac{5 \pi}{6}\).

Use the sine property of complementary angles

The sine function has the property that \(\sin(\pi - \theta) = \sin(\theta)\). Using this property, we can rewrite the angles as follows: \(\sin \frac{2 \pi}{3} = \sin (\pi - \frac{\pi}{3}) = \sin \frac{1 \pi}{3}\) \(\sin \frac{5 \pi}{6} = \sin (\pi - \frac{\pi}{6}) = \sin \frac{1 \pi}{6}\) Now, we can simplify our given expression as: \(\sin \frac{1 \pi}{6} \sin \frac{1 \pi}{3} \sin \frac{2 \pi}{3} \sin \frac{5 \pi}{6} = \sin \frac{1 \pi}{6} \sin \frac{1 \pi}{3} \sin \frac{1 \pi}{3} \sin \frac{1 \pi}{6}\)

Substitute the values

We can now substitute the values for \(\sin \frac{\pi}{6}\) and \(\sin \frac{\pi}{3}\) from step 2: \(\sin \frac{1 \pi}{6} \sin \frac{1 \pi}{3} \sin \frac{1 \pi}{3} \sin \frac{1 \pi}{6} = \frac{1}{2}\cdot\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{3}}{2}\cdot\frac{1}{2}\)

Solve the expression

Multiply the fractions: \(\frac{1}{2}\cdot\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{3}}{2}\cdot\frac{1}{2} = \frac{1\cdot\sqrt{3}\cdot\sqrt{3}\cdot 1}{2\cdot 2\cdot 2\cdot 2} = \frac{3}{16}\) Comparing with the given choices, we can see that the correct option is (b). Therefore, the value of \(\sin \frac{\pi}{6} \sin \frac{2 \pi}{6} \sin \frac{4 \pi}{6} \sin \frac{5 \pi}{6}\) is \(\boxed{\frac{3}{16}}\).

Key concepts

Trigonometric Functions

Trigonometric functions are essential building blocks in mathematics, often used to describe relationships in a right triangle. There are six main functions: sine (\( \sin \theta \)), cosine (\( \cos \theta \)), tangent (\( \tan \theta \)), cosecant (\( \csc \theta \)), secant (\( \sec \theta \)), and cotangent (\( \cot \theta \)). Each of these functions provides a ratio from a right triangle's sides.
The focus here is on the sine function, which relates the opposite side to the hypotenuse of a right triangle. It's important because it helps to read various angles and quickly retrieve their relationship between the geometric properties. Knowing these can help solve both theoretical and practical problems related to angles and circles.
An easy way to remember these trigonometric functions is to use mnemonic devices. For instance, SOH-CAH-TOA hints at:

• SOH: Sine is Opposite over Hypotenuse
• CAH: Cosine is Adjacent over Hypotenuse
• TOA: Tangent is Opposite over Adjacent

This understanding sets a foundation for approaching any problem involving angles and sides of triangles.

Unit Circle

The unit circle is a powerful tool in understanding trigonometric functions. The unit circle is simply a circle with a radius of 1, centered at the origin of the coordinate plane. The circle itself is a complete representation of all possible angles a line segment from the center can make with the x-axis.
Each point along the unit circle can be expressed using trigonometric functions. For any angle \( \theta \), the coordinates of the point on the circle are (\( \cos \theta \), \( \sin \theta \)). Because the radius is 1, these coordinates seamlessly translate between the geometric and algebraic representations of trigonometric values.
The unit circle allows mathematicians to extend trigonometric functions beyond 90 degrees, providing consistency across all angles through symmetry and periodicity. You can also use it easily to convert angles between radians and degrees, facilitating better understanding and computation of trigonometric values.

Complementary Angles

Complementary angles are two angles whose sum is \( \pi/2 \) radians or 90 degrees. Trigonometric functions of complementary angles have particular properties that are helpful when solving problems.
One of the key identities involves the sine and cosine functions: \( \sin \theta = \cos(\pi/2 - \theta) \) and \( \cos \theta = \sin(\pi/2 - \theta) \). These relationships make it possible to transform problems involving certain angles into equivalent problems that involve simpler, or more known, angles.
In the provided exercise, recognizing complementary angles simplifies the problem remarkably. Instead of directly calculating \( \sin \frac{2\pi}{3} \), using the property, we convert it to \( \sin \frac{\pi}{3} \). Such conversions help to utilize known sine and cosine values for efficient problem solving.

Radian Measure

Radian measure is a different way of expressing angles, often resulting in more streamlined computations within calculus. Rather than using degrees, radians express angles in terms of the distance around the unit circle.
There is a direct conversion between degrees and radians: 180 degrees is equivalent to \( \pi \) radians. Therefore, to convert degrees to radians, multiply by \( \frac{\pi}{180} \). For instance, 30 degrees become \( \frac{\pi}{6} \) radians.
Radians offer a more natural integration into mathematical functions, as they establish a direct relationship with circle radius and arc length. Most notably, they allow trigonometric functions to align with calculus operations seamlessly, as derivatives and integrals of these functions use radians to maintain consistency with their geometric interpretations.
In trigonometry and calculus, working in radian measure can simplify algebra and improve intuition about angle measures. This practice is crucial when dealing with sine, cosine, and other trigonometric tasks, as values and functions are universally tabulated in radians.