已知sinθ+cosθ=√2 /3,(0
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已知sinθ+cosθ=√2 /3,(0
![已知sinθ+cosθ=√2 /3,(0](/uploads/image/z/4594835-11-5.jpg?t=%E5%B7%B2%E7%9F%A5sin%CE%B8%2Bcos%CE%B8%3D%E2%88%9A2+%2F3%2C%280)
∵ sin θ + cos θ = √2 / 3
∴ (sin θ + cos θ)² = 2 / 9
sin ² θ + 2 sin θ cos θ + cos ² θ = 2 / 9
∵ sin ² θ + cos ² θ = 1
∴ 1 + 2 sin θ cos θ = 2 / 9
2 sin θ cos θ = - 7 / 9
sin θ cos θ = - 7 / 18
∴ sin θ cos θ / 1 = - 7 / 18
∴ sin θ cos θ / (sin ² θ + cos ² θ) = - 7 / 18
分子分母同除以 cos ² θ 得:
tan θ / (tan ² θ + 1)= - 7 / 18
tan θ = (- 7 / 18)(tan ² θ + 1)
两边除以 - 7 / 18 得:
(- 18 / 7)tan θ = tan ² θ + 1
tan ² θ + (18 / 7)tan θ + 1 = 0
tan ² θ + (18 / 7)tan θ + (9 / 7)² = - 1 + (9 / 7)²
(tan θ + 9 / 7)² = - 1 + 81 / 49
(tan θ + 9 / 7)² = 32 / 49
tan θ + 9 / 7 = ± 4√2 / 7
tan θ = ± 4√2 / 7 - 9 / 7
∴ tan θ = (4√2 - 9) / 7 或 (- 4√2 - 9)/ 7
∵ sin θ cos θ = - 7 / 18 < 0 ,0 < θ < π
∴ π / 2 < θ < π
∴ tan θ < 0
∴ tan θ = (- 4√2 - 9)/ 7
∴ (sin θ + cos θ)² = 2 / 9
sin ² θ + 2 sin θ cos θ + cos ² θ = 2 / 9
∵ sin ² θ + cos ² θ = 1
∴ 1 + 2 sin θ cos θ = 2 / 9
2 sin θ cos θ = - 7 / 9
sin θ cos θ = - 7 / 18
∴ sin θ cos θ / 1 = - 7 / 18
∴ sin θ cos θ / (sin ² θ + cos ² θ) = - 7 / 18
分子分母同除以 cos ² θ 得:
tan θ / (tan ² θ + 1)= - 7 / 18
tan θ = (- 7 / 18)(tan ² θ + 1)
两边除以 - 7 / 18 得:
(- 18 / 7)tan θ = tan ² θ + 1
tan ² θ + (18 / 7)tan θ + 1 = 0
tan ² θ + (18 / 7)tan θ + (9 / 7)² = - 1 + (9 / 7)²
(tan θ + 9 / 7)² = - 1 + 81 / 49
(tan θ + 9 / 7)² = 32 / 49
tan θ + 9 / 7 = ± 4√2 / 7
tan θ = ± 4√2 / 7 - 9 / 7
∴ tan θ = (4√2 - 9) / 7 或 (- 4√2 - 9)/ 7
∵ sin θ cos θ = - 7 / 18 < 0 ,0 < θ < π
∴ π / 2 < θ < π
∴ tan θ < 0
∴ tan θ = (- 4√2 - 9)/ 7
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