Question

Найти sinx*cosx если cosx=7sinx

Answer 1

$cosx=7sinx\\ =>1=7tgx=>tgx=\frac{1}{7}$

$sinx*cosx=\frac{1}{2}*2*sinx*cosx=\frac{1}{2}sin2x$

$po formule\\ sin2x=\frac{2tgx}{1+tg^2x}$

$=> \frac{1}{2}sin2x=\frac{1}{2}*\frac{2tgx}{1+tg^2x}=\\ \frac{1}{2}*\frac{2*\frac{1}{7}}{1+\frac{1}{49}}=\frac{7}{50}=\frac{14}{100}=0.14$

Answer 2

To find the value of sin(x)*cos(x) given that cos(x) = 7sin(x), we can use trigonometric identities to express sin(x)*cos(x) in terms of one variable. Let's proceed step by step:

1. Start with the given equation: cos(x) = 7sin(x).

2. Recall the Pythagorean identity: sin^2(x) + cos^2(x) = 1.

3. Square both sides of the given equation to eliminate the trigonometric functions: cos^2(x) = (7sin(x))^2 cos^2(x) = 49sin^2(x)

4. Now, use the Pythagorean identity to replace sin^2(x) in terms of cos^2(x): 49sin^2(x) = 49(1 - cos^2(x))

5. Rearrange the equation to solve for cos^2(x): 49 - 49cos^2(x) = 49sin^2(x)

6. Substitute the given equation cos(x) = 7sin(x) into the above equation: 49 - 49cos^2(x) = 49(7sin(x))^2 49 - 49cos^2(x) = 49(49sin^2(x))

7. Use the fact that sin^2(x) = (1 - cos^2(x)) (from the Pythagorean identity) to simplify further: 49 - 49cos^2(x) = 49(49 - cos^2(x))

8. Distribute on the right-hand side: 49 - 49cos^2(x) = 2401 - 49cos^2(x)

9. Now, move all the terms involving cos^2(x) to one side of the equation: 49cos^2(x) - 49cos^2(x) = 2401 - 49

10. Simplify the equation: 0 = 2352

The equation simplifies to 0 = 2352, which is not a true statement. This implies that there is no valid solution for x that satisfies the given equation cos(x) = 7sin(x). Therefore, sin(x)*cos(x) cannot be determined with the given information.