2sec(11)sec(19)-2cot(71)
I notice that
cos(30) = cos(11+19)
cos30 = cos11cos19 - sin11sin19
√3/2 + sin11sin19 = cos11cos19
and cot71 = cos71/sin71
= cos(60+11)/sin(60+11)
= (cos60cos11 - sin60sin11)/sin60cos11 + cos60sin11)
= (1/2 cos11 - √3/2 sin11)/(√3/2 cos11 + 1/2 sin11)
= (cos11 - √3sin11)/(√3cos11 + sin11)
2sec(11)sec(19)-2cot(71)
= 2/(cos11cos19) - 2cot71
= 2/(√3/2 + sin11sin19 ) - 2(cos11 - √3sin11)/(√3cos11 + sin11)
= 4/(√3 + 2sin11sin19) - 2(cos11 - √3sin11)/(√3cos11 + sin11)
At this point I don't see my way out of this quagmire, but I realize that 11+19 - 30, perhaps somebody else can see the light from here.
To evaluate the expression 2sec(11)sec(19)-2cot(71), we need to understand the trigonometric functions involved.
1. Secant function (sec): The secant of an angle is equal to the reciprocal of the cosine of that angle. Mathematically, sec(x) = 1/cos(x).
2. Cotangent function (cot): The cotangent of an angle is equal to the reciprocal of the tangent of that angle. Mathematically, cot(x) = 1/tan(x).
Now, let's calculate each term step by step:
1. sec(11):
To find sec(11), we need to evaluate 1/cos(11). This can be done using a scientific calculator or a trigonometric function table. If you're using a scientific calculator, the steps are as follows:
- Make sure your calculator is set to degrees mode.
- Calculate the cosine of 11 degrees: cos(11) ≈ 0.988.
- Take the reciprocal to find sec(11): sec(11) ≈ 1/0.988 ≈ 1.012.
2. sec(19):
Similar to sec(11), we need to evaluate 1/cos(19). Using the steps mentioned above, if we calculate cos(19), we find cos(19) ≈ 0.947. Taking the reciprocal, we get sec(19) ≈ 1/0.947 ≈ 1.056.
3. cot(71):
To find cot(71), we need to evaluate 1/tan(71). Again, using a scientific calculator or a trigonometric function table, the steps are:
- Make sure your calculator is set to degrees mode.
- Calculate the tangent of 71 degrees: tan(71) ≈ 2.495.
- Take the reciprocal to find cot(71): cot(71) ≈ 1/2.495 ≈ 0.401.
Now, we can substitute the calculated values back into the original expression:
2sec(11)sec(19)-2cot(71) ≈ 2(1.012)(1.056)-2(0.401)
Performing the multiplication:
≈ 2.14 - 0.802
Finally, subtracting the terms:
≈ 1.338
Therefore, the value of the expression 2sec(11)sec(19)-2cot(71) is approximately 1.338.