Question

Sin α= 5/13 , π/2<α<π Найти sin 2α, cos 2α, tg 2α. C объяснением

Answer 1

Ответ:

cos^2 (a) + sin^2(a) = 1

cos^2(a) = 169/169 - 25/169 = 144/169

cos(a) = - 12/13 ( т. к. 3я четверть)

sin(2a) = 2sin(a) * cos(a) = 2 * 12/13 * 5/13 = 120/169

Answer 2

To find the values of sin 2α, cos 2α, and tan 2α, we can use the double angle formulas for trigonometric functions. For any angle α, these formulas are:

1. sin 2α = 2 * sin α * cos α
2. cos 2α = cos^2 α - sin^2 α
3. tan 2α = (2 * tan α) / (1 - tan^2 α)

Given that sin α = 5/13 and π/2 < α < π, we know that sin α is positive (since it's in the second quadrant). To find cos α, we can use the Pythagorean identity: sin^2 α + cos^2 α = 1.

1. sin^2 α + cos^2 α = 1 (5/13)^2 + cos^2 α = 1 25/169 + cos^2 α = 1 cos^2 α = 1 - 25/169 cos^2 α = 144/169 cos α = √(144/169) cos α = 12/13 (since cos α is positive in the second quadrant)

Now, we can find the values of sin 2α, cos 2α, and tan 2α using the formulas above:

1. sin 2α = 2 * sin α * cos α sin 2α = 2 * (5/13) * (12/13) sin 2α = 120/169

2. cos 2α = cos^2 α - sin^2 α cos 2α = (12/13)^2 - (5/13)^2 cos 2α = 144/169 - 25/169 cos 2α = 119/169

3. tan 2α = (2 * tan α) / (1 - tan^2 α) tan 2α = (2 * (5/12)) / (1 - (5/12)^2) tan 2α = (10/12) / (1 - 25/144) tan 2α = (5/6) / (119/144) tan 2α = (5/6) * (144/119) tan 2α = 20/17

So, the values are:

• sin 2α = 120/169
• cos 2α = 119/169
• tan 2α = 20/17