Question

Найти sin a + cos a, если sin 2a = 0.84, 3П/4 < a < П

Answer 1

В четвёртой четверти: sin a < 0, cos a > 0.

Тогда найдём искомую сумму через квадратное уравнение: (sin a + cos a)^2=sin^2(a)+2sin a*cos a+cos^2(a)=1+2sinacosa=1+sin2a=1+0,84=1,84 => sin a + cos a =sqrt(1,84)=1,36.

Answer 2

To find the value of sin(a) + cos(a) given the information that sin(2a) = 0.84 and 3π/4 < a < π, we can use trigonometric identities and the given information.

First, let's use the double angle identity for sine:

sin(2a) = 2 * sin(a) * cos(a)

Since sin(2a) = 0.84, we can write:

0.84 = 2 * sin(a) * cos(a)

Now, we also know that 3π/4 < a < π. This means a is in the second quadrant where both sin(a) and cos(a) are negative.

Now, we need to find the values of sin(a) and cos(a) using the given information.

1. From the inequality 3π/4 < a < π, we can tell that a is in the second quadrant, where cos(a) is negative.

2. We can rearrange the equation we derived earlier:

sin(a) * cos(a) = 0.84 / 2

sin(a) * cos(a) = 0.42

3. Since sin(a) and cos(a) are both negative in the second quadrant, we can write:

sin(a) = -√(1 - cos²(a))

Now, we can substitute this into the equation we derived in step 2:

-√(1 - cos²(a)) * cos(a) = 0.42

Solve for cos(a):

1 - cos²(a) = (0.42 / cos(a))²

1 - cos²(a) = 0.1764 / cos²(a)

Multiply through by cos²(a):

cos²(a) - cos⁴(a) = 0.1764

Let y = cos²(a):

y - y² = 0.1764

Rearrange as a quadratic equation:

y² - y + 0.1764 = 0

Now, we can solve this quadratic equation for y (cos²(a)).

Using the quadratic formula: y = [-b ± √(b² - 4ac)] / 2a

where a = 1, b = -1, and c = 0.1764

y = [1 ± √(1 - 4 * 1 * 0.1764)] / 2

y = [1 ± √(1 - 0.7056)] / 2

y = [1 ± √0.2944] / 2

y = [1 ± 0.5429] / 2

Now, we have two potential values for y:

1. y = (1 + 0.5429) / 2 = 1.5429 / 2 ≈ 0.7715
2. y = (1 - 0.5429) / 2 = 0.4571 / 2 ≈ 0.2286

However, since cos(a) is negative in the second quadrant, the second value (y ≈ 0.2286) is not applicable. Thus, we take the first value (y ≈ 0.7715).

cos²(a) ≈ 0.7715

Now, take the square root of both sides:

cos(a) ≈ √0.7715 ≈ 0.8783

Now that we have cos(a), we can find sin(a):

sin(a) = -√(1 - cos²(a))

sin(a) ≈ -√(1 - 0.7715) ≈ -√0.2285 ≈ -0.4780

Finally, to find sin(a) + cos(a):

sin(a) + cos(a) ≈ -0.4780 + 0.8783 ≈ 0.4003

So, sin(a) + cos(a) ≈ 0.4003.