Question

Sin(-71п/4) : cos(-101п/3)

Answer 1

Используем формулы: sin(-α)=-sinα; cos(-α)=cosα;
Представим углы в виде сумм: 71П/4=18П-П/4  и 101П/3=32П-П/3. 
Также используем формулы синуса разности и косинуса разности:
sin(α-β)= sinα·cosβ-cosα·sinβ, cos(α-β)=cosα·cosβ+sinα·sinβ.
(sin(-71П/4)) / (cos(-101П/3)=-(sin(71П/4)) / (cos(101П/3)=
=-(sin(18П-П/4)) / (cos(34П-П/3)=
=-(sin(18П)·cos(П/4)-cos(18П)·sin(П/4)) / (cos(25П)·cos(П/3)-sin(25П)·sin(П/3))=
=(0·(√2/2)-1·(√2/2)) / ((-1)·(1/2)-0·(1/√3))=(-√2/2)/(-1/2)=√2

Answer 2

To evaluate the trigonometric expressions, let's first convert the angles from radians to degrees:

1. Convert -71π/4 radians to degrees: -71π/4 radians = (-71/4) * π radians ≈ -17.6777 radians -17.6777 radians ≈ -17.6777 * (180/π) degrees ≈ -1012.5 degrees

2. Convert -101π/3 radians to degrees: -101π/3 radians = (-101/3) * π radians ≈ -33.5113 radians -33.5113 radians ≈ -33.5113 * (180/π) degrees ≈ -1921.3 degrees

Now, let's evaluate the trigonometric expressions:

1. Sin(-71π/4): Sin(-71π/4) = Sin(-1012.5 degrees)

Since the sine function is periodic with a period of 360 degrees, we can simplify the angle by adding or subtracting multiples of 360 degrees:

Sin(-1012.5 degrees) = Sin(-1012.5 degrees + 2 * 360 degrees) = Sin(247.5 degrees)

Sin(247.5 degrees) = -1 (since sine is -1 at 247.5 degrees).

2. Cos(-101π/3): Cos(-101π/3) = Cos(-1921.3 degrees)

Similarly, we can simplify the angle by adding or subtracting multiples of 360 degrees:

Cos(-1921.3 degrees) = Cos(-1921.3 degrees + 5 * 360 degrees) = Cos(58.7 degrees)

However, Cos(58.7 degrees) is not a simple integer value. It is a decimal number.

So, the final result is: Sin(-71π/4) = -1 Cos(-101π/3) ≈ a decimal number (approximately 0.522)