Question

1)tg5x-ctg3x=0 2)sin10x-cos3x=0

Answer 1

Решение на фотографии

Answer 2

1. To solve the equation tg5x - ctg3x = 0, we can use the trigonometric identity:

ctg x = 1 / tan x

So we have:

tg 5x - 1 / tg 3x = 0

Multiplying both sides by tg 3x, we get:

tg 5x tg 3x - 1 = 0

Using the trigonometric identity:

tg a tg b = (sin a sin b) / (cos a cos b)

We can rewrite the equation as:

(sin 5x sin 3x) / (cos 5x cos 3x) - 1 = 0

Multiplying both sides by cos 5x cos 3x, we get:

sin 5x sin 3x - cos 5x cos 3x = 0

Using the trigonometric identity:

cos a sin b - sin a cos b = sin (b - a)

We can rewrite the equation as:

sin (5x - 3x) = 0

Simplifying, we get:

sin 2x = 0

So the solutions are:

x = nπ / 2, where n is an integer.

2. To solve the equation sin 10x - cos 3x = 0, we can use the trigonometric identity:

sin (a - b) = sin a cos b - cos a sin b

So we have:

sin 10x - cos 3x = sin (10x - 90°) - sin (90° + 3x)

Using the trigonometric identity:

sin (a - b) - sin (a + b) = 2 sin a cos b

We can rewrite the equation as:

2 sin (10x - 90° + 3x) cos (10x - 90° - 3x) = 0

Simplifying, we get:

2 sin 13x cos 7x = 0

So the solutions are:

x = nπ / 13 or x = (2n + 1)π / 14, where n is an integer.