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The problem in the image asks for the exact value of \(\sin 30^\circ - \cos 30^\circ\) and provides four options:

a) \(\frac{1 - \sqrt{3}}{2}\)

b) \(\frac{\sqrt{3} - 1}{2}\)

c) \(\frac{3 - \sqrt{3}}{2}\)

d) \(\frac{\sqrt{3}}{2} - \frac{1}{2}\)

Let's calculate \(\sin 30^\circ\) and \(\cos 30^\circ\) to solve this.

1. \(\sin 30^\circ = \frac{1}{2}\)
2. \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)

Now, subtract these values:

\[
\sin 30^\circ - \cos 30^\circ = \frac{1}{2} - \frac{\sqrt{3}}{2} = \frac{1 - \sqrt{3}}{2}
\]

So, the correct answer is:

**a) \(\frac{1 - \sqrt{3}}{2}\)**

Would you like more details on this solution or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the value of \(\cos 60^\circ\) and how does it compare to \(\cos 30^\circ\)?
2. How do the sine and cosine functions behave as angles increase from 0° to 90°?
3. What is the exact value of \(\sin 45^\circ - \cos 45^\circ\)?
4. How do the identities \(\sin^2 \theta + \cos^2 \theta = 1\) and \(\sin(90^\circ - \theta) = \cos \theta\) apply here?
5. Can you derive the sine and cosine values for 30° and 60° using an equilateral triangle?

**Tip:** Remember that the sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse.

Which of the following is the exact value of sin 30° - cos 30°?

Learn how to calculate the exact value of sin 30° - cos 30° with a step-by-step solution. This trigonometry problem is perfect for students in Grades 9-12, covering key concepts like sine and cosine functions and trigonometric identities.

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