Question

1)cosa*cos3a-sina*sin3a 2) sin2a*cosa+cos2a*sina 3) sin40*cos5+cos40*sin5 4) (tg7П/15-tg2П/15)/1+tg7П/15*tg2П/15 5) sina=4/5;П<a<3П/2; cosa-?

Answer 1

1) cos a cos 3a - sin a sin 3a = cos (a+3a) = cos 4a

2) sin 2a cos a + cos 2a sin a = sin (2a+a) = sin 3a

3) sin40°cos5°+cos40°sin5° = sin(40°+5°)= sin 45° = $\frac{\sqrt{2}}{2}$

4) $\dfrac{tg\frac{7\pi}{15}-tg\frac{2\pi}{15}}{1+tg\frac{7\pi}{15}tg\frac{2\pi}{15}} = tg(\frac{7\pi}{15}-\frac{2\pi}{15}) =tg\frac{5\pi}{15}=tg\frac{\pi}{3}=\sqrt{3}$

5) $\pi$

$sin^2\alpha +cos^2\alpha =1\\ cos\alpha =-\sqrt{1-(\frac{4}{5})^2} =-\sqrt{1-\frac{16}{25}}=-\sqrt{\frac{9}{25}}=-\frac{3}{5}$

Answer 2

1. To simplify the expression cosacos3a - sinasin3a, we can use the trigonometric identity for the cosine of the difference of angles:

cosacos3a - sinasin3a = cos(a - 3a) = cos(-2a) = cos(2a)

So the simplified expression is cos(2a).

2. Similarly, for the expression sin2acosa + cos2asina, we can use the trigonometric identity for the sine of the sum of angles:

sin2acosa + cos2asina = sin(2a + a) = sin(3a)

So the simplified expression is sin(3a).

3. For the expression sin40cos5 + cos40sin5, we can rewrite it using the trigonometric identity for the sine of the sum of angles:

sin(40 + 5) = sin(45)

Since sin(45) is a known value, the simplified expression is sin(45) = 1/√2 or approximately 0.707.

4. The expression (tg(7π/15) - tg(2π/15)) / (1 + tg(7π/15) * tg(2π/15)) can be simplified using the trigonometric identity for tangent of the difference of angles:

tg(7π/15) - tg(2π/15) = tg(7π/15 - 2π/15) = tg(5π/15) = tg(π/3) = √3

Similarly, using the trigonometric identity for tangent of the sum of angles:

1 + tg(7π/15) * tg(2π/15) = 1 + tg(7π/15 + 2π/15) = 1 + tg(9π/15) = 1 + tg(3π/5) = 1 + √3

Therefore, the simplified expression is (√3) / (1 + √3).

5. Given that sina = 4/5 and π < a < (3π/2), we can determine the value of cosa using the Pythagorean identity:

cos^2(a) + sin^2(a) = 1

Replacing sin(a) with 4/5, we get:

cos^2(a) + (4/5)^2 = 1 cos^2(a) + 16/25 = 1 cos^2(a) = 1 - 16/25 cos^2(a) = 9/25

Taking the square root of both sides:

cos(a) = ±√(9/25) = ±3/5

Since a is in the range π < a < (3π/2), sin(a) is positive and cos(a) is negative. Therefore, cos(a) = -3/5.