Problem 58 - Precalculus: Functions and Graphs

Question

Show that the equation is not an Identity. $$\sin (t+\pi)=\sin t$$

Step-by-Step Solution

Answer

The equation \( \sin(t + \pi) = \sin t \) is not an identity because it simplifies to \( \sin(t + \pi) = -\sin t \).

Step 1: Recall the Sine Addition Formula

The identity for the sine of a sum can be given by the formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). In this exercise, we will use \( a = t \) and \( b = \pi \).

Step 2: Apply the Sine Addition Formula

Substitute \( a = t \) and \( b = \pi \) into the sine addition formula. We get \( \sin(t + \pi) = \sin t \cos \pi + \cos t \sin \pi \).

Step 3: Evaluate Trigonometric Values

Recall that \( \cos \pi = -1 \) and \( \sin \pi = 0 \). Substitute these values into the equation: \( \sin(t + \pi) = \sin t \times (-1) + \cos t \times 0 \).

Step 4: Simplification

After substitution, the equation simplifies to \( \sin(t + \pi) = -\sin t \).

Step 5: Compare Both Sides of the Equation

The original equation \( \sin(t + \pi) = \sin t \) is not equivalent to \( \sin(t + \pi) = -\sin t \), thus showing they are not identical. Therefore, the given equation is not an identity.

Key Concepts

Sine Addition Formula

The sine addition formula is a fundamental identity in trigonometry that combines the sine of two angles into a single expression. It states:

• \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

This formula is especially useful when dealing with the sine of angles that are sums of other angles. In many trigonometric problems, breaking down complex expressions into simpler trigonometric components with this formula aids in finding solutions. For instance, in our problem, we applied it with \( a = t \) and \( b = \pi \). Understanding this formula allows us to transform and simplify trigonometric expressions effectively.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to the ratios of the sides of a right triangle. There are six main trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. Each of these functions has its own unique properties and applications. But they are all interconnected through various identities and formulas.

• Sine (\( \sin \)) represents the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)) is the ratio of the sine of an angle to the cosine of that angle.

These functions are used to model periodic phenomena such as sound and light waves, making them crucial in many fields including engineering and physics.

Sine Function

The sine function, noted as \( \sin \), is a periodic function with a wave-like pattern. It is important for its role in the unit circle. For any angle \( t \), \( \sin t \) gives the y-coordinate of the point where the terminal side of the angle intercepts the unit circle. A few key features of the sine function are:

• The function oscillates between -1 and 1.
• It has a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.
• The sine function is an odd function, so \( \sin(-t) = -\sin t \).

These properties make the sine function incredibly versatile for expressing oscillatory motion and understanding wave behavior.

Equivalence of Trigonometric Expressions

Determining whether two trigonometric expressions are equivalent is an important aspect of solving trigonometric problems. Two expressions are equivalent if they yield the same values for all permissible values of the variable.In the exercise, we investigated the supposed identity \( \sin(t + \pi) = \sin t \). By using the sine addition formula, we rewrote the left-hand side:

• \( \sin(t + \pi) = \sin t \cos \pi + \cos t \sin \pi \).
• Given \( \cos \pi = -1 \) and \( \sin \pi = 0 \), this simplifies to \( -\sin t \).

Thus showing that \( \sin(t + \pi) = \sin t \) is not true for all \( t \) and is not a trigonometric identity.