The value of the expression \(\sin 80^\circ-\cos 80^\circ\) is negative.
False.
Let us carefully check step by step:
Step 1: Recall the meaning of sine and cosine values.
- For any angle \(\theta\) (measured in degrees here), \(\sin \theta\) and \(\cos \theta\) are numbers between 0 and 1 (in SI system, they are pure ratios, so they do not have any units).
Step 2: Compare sine and cosine in the range \(45^\circ \leq \theta \leq 90^\circ\).
- At \(45^\circ\), both are equal: \(\sin 45^\circ = \cos 45^\circ = 0.707\).
- For angles larger than \(45^\circ\), the sine value keeps increasing towards 1, while the cosine value keeps decreasing towards 0.
Step 3: At \(\theta = 80^\circ\):
- \(\sin 80^\circ \approx 0.985\)
- \(\cos 80^\circ \approx 0.173\)
Step 4: Subtract the two values:
\[
\sin 80^\circ - \cos 80^\circ = 0.985 - 0.173 = 0.812
\]
Step 5: The result (0.812) is positive, not negative.
Final Conclusion: The given statement is False. The expression \(\sin 80^\circ - \cos 80^\circ\) is positive.