Problem 11 - Calculus Single Variable

Question

Calculate the given expression without using a calculator. \(\sin (\pi \cdot \sin (\pi / 6))\)

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Calculate \( \sin (\pi / 6) \)

To solve the expression, start by finding \( \sin (\pi / 6) \). From trigonometry, we know that \( \sin (\pi / 6) = 1/2 \).

Step 2: Substitute \( \sin (\pi / 6) \) into \( \sin (\pi \cdot x) \)

Now, substitute the value found in Step 1 into the expression \( \sin (\pi \cdot \sin (\pi / 6)) \). This becomes \( \sin (\pi \cdot 1/2) \).

Step 3: Simplify the Expression \( \sin (\pi / 2) \)

The expression \( \sin (\pi / 2) \) corresponds to the sine of 90 degrees, which is well known to be 1.

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It relates the ratios of sides in a right triangle to an angle's sine. For any angle \( \theta \), the sine function \( \sin(\theta) \) gives the ratio of the length of the side opposite the angle to the hypotenuse in the right triangle where the angle \( \theta \) is placed. This being a periodic function, it repeats its values in a regular interval. In particular, sine is periodic with a period of \( 2\pi \) radians or 360 degrees.

• Sine values range between -1 and 1.
• It is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \).
• The sine of significant common angles is often memorized to assist in calculations, such as \( \sin(\pi/6) = 1/2 \) and \( \sin(\pi/2) = 1 \).

Understanding these properties is critical for solving expressions involving the sine function, such as computing \( \sin(\pi \cdot \frac{1}{2}) = \sin(\pi/2) \).

Radians

Radians offer an alternative to degrees for measuring angles, especially in mathematics and trigonometry. An angle's measurement in radians is based on the radius of a circle, corresponding to the length of the arc that the angle subtends.

• There are \( 2\pi \) radians in a full circle, equivalent to 360 degrees.
• This relationship means \( \pi \) radians equals 180 degrees.
• Common conversions used are: \( \pi/6 \) radians equals 30 degrees and \( \pi/2 \) radians equals 90 degrees.

Radians naturally appear in many mathematical formulas and identities, making them convenient for theoretical work. In our problem, converting angles to radians simplifies the use of trigonometric functions and their periodic properties.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true for all angles. They serve as handy tools when simplifying expressions or solving trigonometric equations. For example:

• Reciprocal identities: relating sine to cosecant: \( \csc(\theta) = 1/\sin(\theta) \).
• Pythagorean identities, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Angle sum/difference identities to find the sine or cosine of combined angles.

While these are just a few examples, each identity supports solving complex trigonometric problems efficiently. Knowing identities like \( \sin(\pi/2) = 1 \), simplifies calculations in our problem, assisting us directly to find values exactly without needing a calculator.