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)cosa*cos3a-sina*sin3a =cos(a+3a)=cos4a
2) sin2a*cosa+cos2a*sina = sin(2a+a)=sin3a
3) sin40*cos5+cos40*sin5 =sin(40+5)=sin45=√2/2
4) (tg7П/15-tg2П/15)/1+tg7П/15*tg2П/15=tg(7П/15-2П/15)=tg(п/3)=√3
5) sina=-4/5;П<a<3П/2; cosa-?
cosa на отрезке П<a<3П/2 отрицателен.
1-sin^2a=cos^2a=1-16/25=9/25
cosa=-3/5

Answer 2

Let's solve each of the given trigonometric expressions step by step:

1. To simplify the expression cosa*cos3a - sina*sin3a, we can use the cosine and sine angle addition formulas:

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cosa*cos3a - sina*sin3a = cos(a + 2a) - sin(a + 2a)
                       = cos(3a) - sin(3a)

The values of cos(3a) and sin(3a) can be found using the triple-angle formulas:

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cos(3a) = 4*cos^3(a) - 3*cos(a)
sin(3a) = 3*sin(a) - 4*sin^3(a)

Substituting these values into the expression:

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cos(3a) - sin(3a) = 4*cos^3(a) - 3*cos(a) - (3*sin(a) - 4*sin^3(a))

2. The expression sin2a*cosa + cos2a*sina can be simplified using the double-angle formulas:

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sin2a = 2*sin(a)*cos(a)
cos2a = cos^2(a) - sin^2(a) = cos^2(a) - (1 - cos^2(a)) = 2*cos^2(a) - 1

Substituting these values into the expression:

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sin2a*cosa + cos2a*sina = 2*sin(a)*cos(a)*cos(a) + (2*cos^2(a) - 1)*sin(a)

3. For the expression sin40*cos5 + cos40*sin5, we can't simplify it further without knowing the specific values of sin(40) and cos(40). Please provide the actual values of sin(40) and cos(40) if you have them.

4. The expression (tg(7π/15) - tg(2π/15)) / (1 + tg(7π/15)*tg(2π/15)) involves tangent functions. To simplify it, we can use the tangent subtraction formula:

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tg(a - b) = (tg(a) - tg(b)) / (1 + tg(a)*tg(b))

Substituting a = 7π/15 and b = 2π/15:

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tg(7π/15 - 2π/15) = (tg(7π/15) - tg(2π/15)) / (1 + tg(7π/15)*tg(2π/15))

5. Given sin(a) = 4/5 and π < a < (3π/2), we know that a lies in the third quadrant where both sine and cosine are negative. Using the Pythagorean identity sin^2(a) + cos^2(a) = 1, we can find cos(a):

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cos^2(a) = 1 - sin^2(a) = 1 - (4/5)^2 = 1 - 16/25 = 9/25
cos(a) = sqrt(9/25) = 3/5 (taking the negative value since `a` is in the third quadrant)

Please note that the solution to the third expression requires the specific values of sin(40) and cos(40) to proceed further.