Question

Sin50sin20+cos20sin40/cos40cos70+sin70cos50

Answer 1

-0.26*0.91+0.41*0.64/(-0.67)*0.63+0.94*0.96=

= - 0.24+0.26/(-0.42)+0.90= 0.02/0.48.
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Answer 2

To simplify the given trigonometric expression, let's break it down step by step:

Step 1: Convert all angles to degrees (assuming they are in degrees, not radians):

• sin(50°) * sin(20°) + cos(20°) * sin(40°) / cos(40°) * cos(70°) + sin(70°) * cos(50°)

Step 2: Use trigonometric identities to simplify the expression:

• sin(a) * sin(b) = 1/2 * [cos(a - b) - cos(a + b)]
• cos(a) * sin(b) = 1/2 * [sin(a + b) + sin(a - b)]
• cos(a) * cos(b) = 1/2 * [cos(a + b) + cos(a - b)]

Applying these identities, the expression becomes:

• 1/2 * [cos(50° - 20°) - cos(50° + 20°)] + 1/2 * [sin(20° + 40°) + sin(20° - 40°)] / 1/2 * [cos(40° + 70°) + cos(40° - 70°)] + 1/2 * [sin(70° + 50°) + sin(70° - 50°)]

Step 3: Simplify further:

• 1/2 * [cos(30°) - cos(70°)] + 1/2 * [sin(60°) + sin(-20°)] / 1/2 * [cos(110°) + cos(-30°)] + 1/2 * [sin(120°) + sin(20°)]

Step 4: Evaluate the trigonometric functions for each angle:

• cos(30°) = sqrt(3)/2
• cos(70°) ≈ 0.342
• sin(60°) = sqrt(3)/2
• sin(-20°) = -sin(20°) = -0.342
• cos(110°) = -cos(70°) ≈ -0.342
• cos(-30°) = cos(30°) = sqrt(3)/2
• sin(120°) = sqrt(3)/2
• sin(20°) ≈ 0.342

Step 5: Substitute the values and calculate the expression:

• 1/2 * [sqrt(3)/2 - 0.342] + 1/2 * [sqrt(3)/2 - 0.342] / 1/2 * [-0.342 + sqrt(3)/2] + 1/2 * [sqrt(3)/2 + 0.342]

Step 6: Perform the arithmetic operations:

• [sqrt(3)/4 - 0.171] + [sqrt(3)/4 - 0.171] / [-0.171 + sqrt(3)/4] + [sqrt(3)/4 + 0.171]

Step 7: Continue simplifying:

• [sqrt(3)/2 - 0.342] / [sqrt(3)/4 - 0.171] + [sqrt(3)/4 + 0.171]

Step 8: Rationalize the denominator by multiplying the numerator and denominator by the conjugate:

• [sqrt(3)/2 - 0.342] * [sqrt(3)/4 + 0.171] / [(sqrt(3)/4 - 0.171) * (sqrt(3)/4 + 0.171)]

Step 9: Simplify the numerator:

• [3/4 - 0.342 * sqrt(3) + 0.171 * sqrt(3) - 0.058482] / [3/16 - (0.171)^2]

Step 10: Combine like terms:

• [3/4 - 0.171 * sqrt(3) + 0.171 * sqrt(3) - 0.058482] / [3/16 - 0.029241]

Step 11: Further simplify:

• [3/4 - 0.058482] / [3/16 - 0.029241]

Step 12: Evaluate the remaining expression:

• [0.941518] / [0.067759]

Step 13: Divide to get the final result:

• ≈ 13.895

So, the simplified value of the given expression is approximately 13.895.