Final answer:
The exact value of tan 202.5° is found by utilizing the periodicity and symmetry of the tangent function, which leads to tan(202.5°) = -tan(22.5°). This uses the fact that the tangent has a period of 180° and is an odd function. Option b is the correct answer.
Step-by-step explanation:
To find the exact value of tan 202.5°, we need to understand the periodicity and symmetry properties of the tangent function. The tangent function has a period of 180°, which means that tan(θ + 180°) = tan(θ).
Moreover, tangent is an odd function, which indicates that tan(-θ) = -tan(θ). Using these properties, tan(202.5°) can be simplified to tan(22.5°) with a negative sign because:
- 202.5° = 180° + 22.5° (which uses the periodicity of tangent)
- tan(180° + θ) = tan(θ) (due to periodicity)
- tan(202.5°) = tan(180° + 22.5°) = tan(22.5°) (applying periodicity)
- Because 202.5° is in the third quadrant where tangent is positive, the actual value is the negation of tan(22.5°).
Therefore, tan(202.5°) = -tan(22.5°), which makes the correct answer b) tan(202.5°) = -tan(22.5°).