Problem 54 - Algebra 2

Question

Use the sum and difference formulas to verify each identity. $$ \sin (\pi-\theta)=\sin \theta $$

Step-by-Step Solution

Answer

The identity \(\sin (\pi-\theta) = \sin \theta\) is verified

Step 1 : Recall the formula for sine of difference of two angles

The sine of difference of two angles \(A\) and \(B\) is given by the formula \(\sin(A - B) = \sin A \cos B - \cos A \sin B \). We will use this formula to solve the exercise.

Step 2 : Apply the sine formula for difference of two angles

Now, apply the formula to \(\sin (\pi-\theta)\). This gives us \(\sin \pi \cos \theta - \cos \pi \sin \theta\). Remember that \(\sin \pi = 0\) and \(\cos \pi = -1\). So, this simplifies to \(0 - (-1) \sin \theta = \sin \theta\). Therefore, we see that \(\sin (\pi-\theta) = \sin \theta\). This completes the verification process.

Key Concepts

Sum and Difference Formulas

In trigonometry, the **sum and difference formulas** are critical tools used to simplify expressions involving trigonometric functions, such as sine and cosine. These formulas help in calculating the trigonometric functions for the sum or difference of two angles. Specifically, for the sine function, the formulas are:

• Sum formula: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
• Difference formula: \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

These identities are derived from the unit circle and can be visualized by considering the coordinates of points on the circle corresponding to given angles. They are useful when dealing with non-standard angles, enabling us to express them as a combination of familiar angles.
Understanding and applying these formulas correctly ensure that you can simplify and solve trigonometric equations efficiently.

Sine Function

The **sine function** is one of the fundamental trigonometric functions. It represents the y-coordinate, or the vertical component, of a point on the unit circle. The function is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) units.
The sine of an angle \(\theta\) can vary from -1 to 1, depending on the angle's position around the circle:

• At \(\theta = 0\), \( \sin \theta = 0 \)
• At \(\theta = \pi/2\), \( \sin \theta = 1 \)
• At \(\theta = \pi\), \( \sin \theta = 0 \)
• At \(\theta = 3\pi/2\), \( \sin \theta = -1 \)
• At \(\theta = 2\pi\), returns to \(0\)

The wave pattern of the sine function is smooth and continuous, making it ideal for describing oscillations in physics, engineering, and other sciences. Recognizing the behavior of the sine function in different quadrants is essential in verifying identities and solving trigonometric equations.

Angle Difference Identities

**Angle difference identities** provide a powerful method for understanding how trigonometric functions relate to each other when considering the difference between angles. These identities break complex angle expressions into simpler, known components. For sine, the angle difference identity is expressed as:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
This identity is nice because it allows direct calculation of the sine of the difference of two angles using the individual sines and cosines of these angles. In the exercise, this identity was used to confirm that \( \sin(\pi - \theta) = \sin \theta \) by plugging in angles \(\pi\) and \(\theta\), and then simplifying based on the known values of sine and cosine for \(\pi\).
By mastering these identities, you gain the ability to solve and manipulate formulas that might initially seem out of reach. They reveal the deep connections within trigonometry, simplifying what can otherwise become intricate calculations. Understanding these principles forms the core of trigonometric problem-solving.