simplify cosec40(sin40+sin80+cos120)
help step
To simplify the expression cosec(40)(sin(40) + sin(80) + cos(120)), we can break it down into smaller parts and simplify each part individually.
Step 1: Calculate sin(40)
To calculate sin(40), we can use a scientific calculator or trigonometric identity. Let's use the calculator approach.
sin(40) ≈ 0.64279 (rounded to five decimal places)
Step 2: Calculate sin(80)
To calculate sin(80), we'll use the calculator approach again.
sin(80) ≈ 0.98481 (rounded to five decimal places)
Step 3: Calculate cos(120)
To calculate cos(120), we'll use the calculator approach.
cos(120) ≈ -0.5 (rounded to one decimal place)
Now that we have the values for sin(40), sin(80), and cos(120), we can simplify the expression.
cosec(40)(sin(40) + sin(80) + cos(120))
= 1/sin(40) * (0.64279 + 0.98481 - 0.5) (substituting the calculated values)
= 1/0.64279 * (1.1276) (simplifying the sum inside the parentheses)
= 1.5549 * 1.1276 (evaluating the reciprocal)
≈ 1.7532 (rounded to four decimal places)
Therefore, the simplified form of the expression cosec(40)(sin(40) + sin(80) + cos(120)) is approximately 1.7532.