Question

Show \(\sin \left(180^{\circ}+\theta\right)=-\sin \theta\).

Short answer

Answer: The value of \(\sin(180^{\circ}+\theta)\) is \(-\sin\theta\).

Step-by-step solution

Convert Degrees to Radians

To work with the angle addition formula for sine, we need to convert the given angle from degrees to radians. We can do this by multiplying the angle in degrees by the conversion factor \(\frac{\pi}{180}\). $$ 180^{\circ} = 180 \cdot \frac{\pi}{180} = \pi $$ Now we can rewrite the given expression as: $$ \sin(\pi + \theta) $$

Apply the Angle Addition Formula for Sine

To solve for \(\sin(\pi + \theta)\), we need to use the angle addition formula for sine, which states: $$ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) $$ In our case, \(\alpha = \pi\) and \(\beta = \theta\). Let's plug these values into the formula: $$ \sin(\pi + \theta) = \sin(\pi)\cos(\theta) + \cos(\pi)\sin(\theta) $$

Use Trigonometric Properties

Now we can use some basic trigonometric properties to simplify the expression further. Recall that \(\sin(\pi) = 0\) and \(\cos(\pi) = -1\). This allows us to simplify the expression as follows: $$ \sin(\pi + \theta) = 0 \cdot \cos(\theta) + (-1) \cdot \sin(\theta) = -\sin(\theta) $$ Now, we have the final result: $$ \sin(180^{\circ}+\theta) = -\sin\theta $$

Key concepts

Radians to Degrees Conversion

Understanding the conversion between radians and degrees is essential in trigonometry. These are two different units for measuring angles. While degrees are more common in everyday use, radians are often preferred in mathematics, especially when dealing with trigonometric functions.

The key conversion factor between degrees and radians is based on the value of a full circle, which is 360 degrees or \(2\pi\) radians. To convert from degrees to radians, multiply the number of degrees by \(\frac{\pi}{180}\).

For example, if you have an angle of \(180^{\circ}\), converting it to radians would involve multiplying by this conversion factor:

• \(180^{\circ} = 180 \times \frac{\pi}{180} = \pi\) radians.

Using this conversion in trigonometric problems allows you to apply standard formulas, like the angle addition formulas, more seamlessly.

Angle Addition Formula

The angle addition formula for sine is a powerful tool in trigonometry. It allows you to find the sine of a sum of two angles. Most importantly, it's expressed as:

\[ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \]

This formula helps in breaking down complex trigonometric expressions into parts that can be handled with basic sine and cosine values. For example, in this exercise, by recognizing \(\alpha = \pi\) and \(\beta = \theta\), using the angle addition formula becomes straightforward.

• \(\sin(\pi + \theta) = \sin(\pi)\cos(\theta) + \cos(\pi)\sin(\theta)\)

Each term is evaluated using known trigonometric properties, simplifying the process and leading us towards the solution.

Trigonometric Properties

Exploring trigonometric properties can greatly simplify calculations. For angle operations, understanding values at pivotal angles like \(\pi\), \(\frac{\pi}{2}\), or \(\pi/2\) is crucial. In this exercise, these properties were used:

• \(\sin(\pi) = 0\)
• \(\cos(\pi) = -1\)

These values are not just arbitrary but stem from the unit circle representation of trigonometric functions.

By substituting these properties back into the expression \(\sin(\pi + \theta)\), it simplifies to \(0 \cdot \cos(\theta) + (-1) \cdot \sin(\theta) = -\sin(\theta)\). Knowing these foundational properties can turn many angle problems into simple arithmetic.