simplify cosec40(sin40+sin80+cos120)
help step
cosec40(sin40+sin80+cos120)
= (1/sin40)(sin40 + sin80 + cos120)
= 1 + sin80/sin40 + cos120/sin40
= 1 + 2sin40cos40/sin40 -1/(2sin40)
= 1 + 2cos40 - (1/2)csc40
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sin80*
To simplify the expression cosec(40)(sin(40) + sin(80) + cos(120)), we can follow the steps below:
Step 1: Calculate the values of sin(40), sin(80), and cos(120).
To do this, we can use a scientific calculator or reference a trigonometric table.
We find:
sin(40) ≈ 0.6428
sin(80) ≈ 0.9848
cos(120) ≈ -0.5
Step 2: Substitute the calculated values back into the expression.
cosec(40)(sin(40) + sin(80) + cos(120))
= cosec(40)(0.6428 + 0.9848 + -0.5)
Step 3: Simplify the expression.
To simplify further, we can add the values inside the parentheses:
0.6428 + 0.9848 + -0.5 ≈ 1.1276
Substituting this back into the expression, we have:
cosec(40)(1.1276)
Step 4: Calculate the value of cosec(40).
cosec(40) is the reciprocal of sin(40), so we can rewrite it as:
1/sin(40)
Using a scientific calculator or reference table, we find:
sin(40) ≈ 0.6428
Therefore,
cosec(40) ≈ 1/0.6428
Calculating this value, we get:
cosec(40) ≈ 1.5557
Step 5: Substitute the value of cosec(40) back into the expression.
cosec(40)(1.1276)
= 1.5557 * 1.1276
Finally, we can calculate this multiplication:
1.5557 * 1.1276 ≈ 1.7542
So, the simplified value of cosec(40)(sin(40) + sin(80) + cos(120)) is approximately 1.7542.