Question

8arcsin^2 x-26arcsin x-7=0

Answer 1

Делаем замену  $arcsinx=t$. 
$8t^{2} -26t-7=0$
]D=676-4*(-7)*8=900
$t_{1} =3,5$
$t_{2} =-0,25$
Делаем обратную замену.  Твердо помним, что арксинус x  это угол синус которого равен x.
$arcsinx=3,5$
3.5 это радианы, 3,5 радиан это  чуть больше  $\pi$ (π≈3,14 радиан), а мы знаем что арксинусы "живут" от $-\frac{ \pi }{2}$  до  $\frac{ \pi }{2}$ 
Значит зесь решений нет.
$arcsinx=-0,25$
 -0.25 радиан попадет в промежуток от $-\frac{ \pi }{2}$  до  $\frac{ \pi }{2}$ 
sin(arcsin x)=sin(-0,25) 
$x=-sin(0,25)$

Answer 2

To solve the equation 8arcsin^2(x) - 26arcsin(x) - 7 = 0, we can make a substitution to simplify the equation. Let's set y = arcsin(x).

Now, we can rewrite the equation in terms of y:

8y^2 - 26y - 7 = 0.

This is a quadratic equation in terms of y. We can solve it using the quadratic formula:

y = (-b ± sqrt(b^2 - 4ac)) / (2a),

where a = 8, b = -26, and c = -7.

Substituting these values into the formula:

y = (-(-26) ± sqrt((-26)^2 - 4 * 8 * (-7))) / (2 * 8) = (26 ± sqrt(676 + 224)) / 16 = (26 ± sqrt(900)) / 16 = (26 ± 30) / 16.

We have two possible solutions:

1. y = (26 + 30) / 16 = 56 / 16 = 7/2.
2. y = (26 - 30) / 16 = -4 / 16 = -1/4.

Now, we substitute back y = arcsin(x) and solve for x.

For y = 7/2: arcsin(x) = 7/2.

Taking the sine of both sides: x = sin(7/2).

For y = -1/4: arcsin(x) = -1/4.

Taking the sine of both sides: x = sin(-1/4).

These are the solutions for the equation 8arcsin^2(x) - 26arcsin(x) - 7 = 0, expressed in terms of the sine function.