Bài tập 6 trang 15 SBT Toán 11 Tập 1 Chân trời sáng tạo - CTST

a) Ta có: sin605° = sin(3.180° + 65°) = ‒ sin65°.

               sin1645° = sin(9.180° + 25°) = ‒ sin25° = ‒ sin(90° ‒ 65°) = ‒ cos65°.

               cot25° = cot(90° ‒ 65°) = tan65°.

               sin²605° + sin²1645° + cot²25° = (‒sin65°)² + (‒cos65°)² + (tan65°)² = 1 + tan²65°= \(\frac{1}{cos^{2}65^{o}}\)

b) Ta có: sin530° = sin(3.180° ‒ 10°) = sin10°.

sin640° = sin(4.180° ‒ 80°) = sin(‒ 80°) = ‒ sin80° = ‒ sin(90° ‒ 10°) = ‒ cos10°.

$\frac{\text{sin}530°}{1+\text{sin}640°}=\frac{\text{sin}10°}{1-\text{cos}10°}=\frac{{\text{sin}}^{2}10°}{\text{sin}10°\left(1-\text{cos}10°\right)}$

$=\frac{1-{\text{cos}}^{2}10°}{\text{sin}10°\left(1-\text{cos}10°\right)}=\frac{\left(1+\text{cos}10°\right)\left(1-\text{cos}10°\right)}{\text{sin}10°\left(1-\text{cos}10°\right)}$

$=\frac{1+\text{cos}10°}{\text{sin}10°}=\frac{1}{\text{sin}10°}+\text{cot}10°\text{ .}$

-- Mod Toán 11 HỌC247