Question

(5sin211+8cos59-5sin31)/(sin54sin67-sin36sin23)​

Answer 1

Ответ:

-2

Объяснение:

(5sin(270-59)+8cos59-5sin(90-59))/(sin(90-36)sin(90-23)-sin36sin23))=

=(-5cod59+8cos59-5cos59)/(cos36cos23-sin36sin23)=

=-2cos59/cos(36+23)=-2cos59/cos59=-2

Answer 2

To simplify the expression, we can use trigonometric identities. Let's start by converting all the trigonometric functions to their respective angle measures:

1. sin(211°) = sin(180° + 31°) = -sin(31°) (sine is negative in the third quadrant)
2. cos(59°) = cos(90° - 31°) = sin(31°) (using complementary angle property)
3. sin(31°) = sin(180° - 149°) = sin(149°) (sine is positive in the second quadrant)
4. sin(54°) = sin(90° - 36°) = cos(36°) (using complementary angle property)
5. sin(67°) = sin(90° - 23°) = cos(23°) (using complementary angle property)
6. sin(36°) and sin(23°) remain as they are.

Now, let's substitute the values into the expression:

(5sin(211°) + 8cos(59°) - 5sin(31°)) / (sin(54°)sin(67°) - sin(36°)sin(23°))

= (5(-sin(31°)) + 8sin(31°) - 5sin(149°)) / (cos(36°)cos(23°) - sin(36°)sin(23°))

= (-5sin(31°) + 8sin(31°) + 5sin(149°)) / (cos(36°)cos(23°) - sin(36°)sin(23°))

= (3sin(31°) + 5sin(149°)) / (cos(36°)cos(23°) - sin(36°)sin(23°))

Now, let's use the following trigonometric identities to simplify further:

1. sin(180° - θ) = sin(θ)
2. sin(θ) = -sin(180° - θ)

Applying these identities:

= (3sin(31°) - 5sin(31°)) / (cos(36°)cos(23°) - sin(36°)sin(23°))

= -2sin(31°) / (cos(36°)cos(23°) - sin(36°)sin(23°))

At this point, we can use another trigonometric identity:

3. sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

Using this identity:

= -2sin(31°) / (sin(36°)sin(23°) - cos(36°)cos(23°))

The denominator is now in the form of sin(A)sin(B) - cos(A)cos(B), which is equivalent to cos(A + B). So we can simplify further:

= -2sin(31°) / cos(36° + 23°)

= -2sin(31°) / cos(59°)

Finally, we can use the identity cos(θ) = sin(90° - θ):

= -2sin(31°) / sin(90° - 59°)

= -2sin(31°) / sin(31°)

The sine of an angle and its complementary angle are equal, so:

= -2

So the simplified value of the expression is -2.