Question

1)3sin²π÷2-4tg²π÷4-3cos²π÷6+3ctg²π÷2 2)tg5π÷4×sinπ÷3-cos7π÷6×sin3π÷2 3)sinα=3÷4,0≤α≤π÷2 4)ctgα=2÷3,π≤α≤3π÷2 5)ctgα-cosα-1÷sinα 6)(sinβ×ctgβ)²+(cosβ×tgβ)2

Answer 1

1)3*1²-4*1²-3*(√3/2)²+3*0²=3-4-2,25+0=-3,25
2)1*√3/2-(-√3/2)*(-1)=√/2-√3/2=0
3)sina=3/4,0≤a≤π/2
a=arcsin3/4
4)ctga=2/3,π≤a3π/2
a=π+arcctg2/3
5)ctga-(cosa-1)/sina=cosa/sina -(cosa -1)/sina=
=(cosa-cosa+1)/sina=1/sina
6)(sinb*ctgb)²+(cosb*tgb)²=(sinb*cosb/sinb)²+(cosb*sinb/cosb)²=
=cos²b+sin²b=1

Answer 2

Let's go through each of the expressions one by one:

1. To simplify the expression 3sin²(π/2) - 4tg²(π/4) - 3cos²(π/6) + 3ctg²(π/2):

• sin²(π/2) equals 1, as sin(π/2) is equal to 1.
• tg(π/4) equals 1, as tg(π/4) or tan(π/4) is equal to 1.
• cos²(π/6) equals 3/4, as cos(π/6) is equal to √3/2, and squaring it gives 3/4.
• ctg(π/2) equals 0, as ctg(π/2) or cot(π/2) is undefined.

Substituting these values into the expression, we have: 3(1) - 4(1) - 3(3/4) + 3(0) = 3 - 4 - (9/4) + 0 = -1 - (9/4) = -13/4

So the simplified value of the expression is -13/4.

2. To simplify the expression tg(5π/4) × sin(π/3) - cos(7π/6) × sin(3π/2):

• tg(5π/4) equals -1, as tg(5π/4) or tan(5π/4) is equal to -1.
• sin(π/3) equals √3/2.
• cos(7π/6) equals √3/2.
• sin(3π/2) equals -1.

Substituting these values into the expression, we have: (-1) × (√3/2) - (√3/2) × (-1) = -√3/2 + √3/2 = 0

So the simplified value of the expression is 0.

3. The equation sin(α) = 3/4, with 0 ≤ α ≤ π/2, means we need to find the value of α within that range that satisfies sin(α) = 3/4.

We can use inverse sine or arcsin to find the angle: α = arcsin(3/4) Using a calculator, we find α ≈ 0.848 radians.

4. The equation ctg(α) = 2/3, with π ≤ α ≤ 3π/2, means we need to find the value of α within that range that satisfies ctg(α) = 2/3.

We can use inverse cotangent or arccot to find the angle: α = arccot(2/3) Using a calculator, we find α ≈ 0.588 radians.

5. To simplify the expression ctg(α) - cos(α) - 1/sin(α):

• ctg(α) equals 1/tan(α), so ctg(α) is the reciprocal of tan(α).
• cos(α) is given as 2/3.
• sin(α) is given as 3/4.

Substituting these values into the expression, we have: 1/tan(α) - 2/3 - 1/(3/4) = 1/tan(α) - 2/3 - 4/3 = 1/tan(α) - 6/3 = 1/tan(α) - 2