Question

Известно, что sint= 5/13 t∈(п/2; п) Найдите sin2t, cos2t, tg2t, ctg2t Задача не с целью просто решить её, а с целью объяснить мне.

Answer 1

$sin^2t+cos^2t=1\\cos^2t=1-sin^2t\\cost=\pm\sqrt{1-sin^2t}$
Т.к. t∈(π/2;π) - 2 четверть, в ней косинус отрицательный. значит перед корнем будет минус.
$cost=-\sqrt{1-(\frac{5}{13})^2}=-\sqrt{\frac{169}{169}-\frac{25}{169}}=-\sqrt{\frac{144}{169}}=-\frac{12}{13}$

$sin2t=2sint*cost=2*\frac{5}{13}*(-\frac{12}{13})=-\frac{120}{169}\\\\cos2t=cos^2t-sin^2t=(-\frac{12}{13})^2-(\frac{5}{13})^2=\frac{144}{169}-\frac{25}{169}=\frac{119}{169}\\\\tg2t=\frac{sin2t}{cos2t}=\frac{-\frac{120}{169}}{\frac{119}{169}}=-\frac{120}{169}*\frac{169}{119}=-\frac{120}{119}\\\\ctg2t=\frac{1}{tg2t}=\frac{1}{-\frac{120}{119}}=-\frac{119}{120}$

Answer 2

Trigonometric Functions and Their Values

To solve this problem, we'll start by understanding the given information. The equation provided is sint = 5/13 and t ∈ (π/2, π). This means that the sine of angle t is 5/13, and the angle t lies in the second quadrant (from π/2 to π).

Finding sin(2t), cos(2t), tan(2t), and cot(2t)

To find sin(2t), cos(2t), tan(2t), and cot(2t), we can use the double-angle identities for trigonometric functions. These identities are:

1. sin(2t) = 2sin(t)cos(t) 2. cos(2t) = cos^2(t) - sin^2(t) 3. tan(2t) = (2tan(t))/(1 - tan^2(t)) 4. cot(2t) = (cot^2(t) - 1)/(2cot(t))

Calculating sin(2t)

First, we'll calculate sin(2t) using the double-angle identity for sine: sin(2t) = 2sin(t)cos(t)

Given that sint = 5/13, we can find cost using the Pythagorean identity: cost = ±sqrt(1 - sin^2(t))

Let's calculate cost: cost = ±sqrt(1 - (5/13)^2) cost = ±sqrt(1 - 25/169) cost = ±sqrt(144/169) cost = ±12/13

Since t is in the second quadrant, cost is negative. Therefore, cost = -12/13.

Now, we can calculate sin(2t): sin(2t) = 2 * (5/13) * (-12/13) sin(2t) = -120/169

So, sin(2t) = -120/169.

Calculating cos(2t)

Next, we'll calculate cos(2t) using the double-angle identity for cosine: cos(2t) = cos^2(t) - sin^2(t)

We already know that sint = 5/13 and cost = -12/13. Now, we can calculate cos(2t): cos(2t) = (-12/13)^2 - (5/13)^2 cos(2t) = 144/169 - 25/169 cos(2t) = 119/169

So, cos(2t) = 119/169.

Calculating tan(2t) and cot(2t)

Finally, we'll calculate tan(2t) and cot(2t) using the identities: tan(2t) = (2tan(t))/(1 - tan^2(t)) cot(2t) = (cot^2(t) - 1)/(2cot(t))

We already know that sint = 5/13 and cost = -12/13. Using these values, we can calculate tan(2t) and cot(2t).

Let's calculate tan(2t): tan(2t) = (2 * (5/13))/ (1 - (5/13)^2) tan(2t) = (10/13)/(1 - 25/169) tan(2t) = (10/13)/(144/169) tan(2t) = (10/13) * (169/144) tan(2t) = 1690/1872

So, tan(2t) = 1690/1872.

Now, let's calculate cot(2t): cot(2t) = ((5/13)^2 - 1)/(2 * (5/13)) cot(2t) = (25/169 - 1)/(10/13) cot(2t) = (-144/169)/(10/13) cot(2t) = -144/130 cot(2t) = -72/65

So, cot(2t) = -72/65.

Conclusion

In conclusion, for the given values of sint = 5/13 and t ∈ (π/2, π), the calculated values are: - sin(2t) = -120/169 - cos(2t) = 119/169 - **tan(2t) =