Question

((sin 10° + sin 80°)(cos 80° - cos 10°)) / sin 70°

Answer 1

Ответ:

-1.

Объяснение:

((sin 10° + sin 80°)(cos 80° - cos 10°)) / sin 70°

sin80° = sin(90°-10°) = cos10°;

cos80° = cos(90°-10°) = sin10°.

Получим, что

((sin 10° + sin 80°)(cos 80° - cos 10°)) / sin 70° = ((sin 10° + cos 10°)( sin 10° - cos 10°)) / sin 70° = (( sin 10°)^2 - (cos 10°)^2) / sin 70° = (-cos(2•10°))/sin 70° = -cos20°/sin70° = -cos20°/sin(90°-20°) = - cos20°/cos20° = - 1.

Answer 2

To calculate the expression ((sin 10° + sin 80°)(cos 80° - cos 10°)) / sin 70°, we can simplify it step by step using trigonometric identities.

Step 1: Expand the numerator ((sin 10° + sin 80°)(cos 80° - cos 10°))

Using the trigonometric identity sin(a) + sin(b) = 2sin((a + b)/2)cos((a - b)/2), we can rewrite the expression as:

2sin(45°)cos(35°) - 2sin(45°)cos(90° - 35°)

Step 2: Simplify the expression 2sin(45°)cos(35°) - 2sin(45°)cos(55°)

Using the trigonometric identity sin(2a) = 2sin(a)cos(a), we can simplify further:

2(sin(45°)cos(35°) - sin(45°)cos(55°))

Step 3: Simplify the denominator sin 70°

Step 4: Substitute the trigonometric values sin(45°) = √2 / 2 cos(35°) = √6 + √2 / 4 cos(55°) = √6 - √2 / 4 sin(70°) = √3 / 2

Step 5: Calculate the expression Substituting the values, we have:

2(√2 / 2)((√6 + √2 / 4) - (√6 - √2 / 4)) / (√3 / 2)

Simplifying further:

2(√2 / 2)(2√2 / 4) / (√3 / 2)

Canceling out common terms:

(√2)(√2) / (√3)

Simplifying the square roots:

2 / (√3)

The final result of the expression ((sin 10° + sin 80°)(cos 80° - cos 10°)) / sin 70° is 2 / (√3).