find the exact value of each expression. do not use a calculator.
Sec 3.14/4 + 2 csc 3.14/3
4 + tan 2*3.14/3
ever hear of pi?
sec pi/4 + 2 csc pi/3
√2 + 2*2/√3
√2 + 4/√3
4+tan 2pi/3
4+(-√3)
4-√3
Using 3.14, you cannot evaluate exactly, even with a calculator.
To find the exact value of each expression, we need to use trigonometric identities and properties. Let's start with the first expression:
Expression 1: sec(3.14/4) + 2csc(3.14/3)
1. Recall the definitions of secant and cosecant functions:
- sec(theta) = 1/cos(theta)
- csc(theta) = 1/sin(theta)
2. Convert the given angles to radians:
- 3.14/4 = 0.7854 radians
- 3.14/3 = 1.0472 radians
3. Compute the values of cosine and sine for the given angles:
- cos(0.7854) = 0.7071
- sin(1.0472) = 0.8660
4. Substitute these values back into the expression:
sec(0.7854) + 2csc(1.0472)
= 1/cos(0.7854) + 2/(1/sin(1.0472))
= 1/0.7071 + 2/0.8660
= 1.4142 + 2.3094
= 3.7236
Therefore, the exact value of expression 1 is 3.7236.
Now let's move on to the second expression:
Expression 2: 4 + tan(2 * 3.14/3)
1. Again, convert the given angle to radians:
- 3.14/3 = 1.0472 radians
2. Compute the value of tangent for the given angle:
- tan(1.0472) = 1.7321
3. Substitute the computed value back into the expression:
4 + tan(1.7321)
= 4 + 1.7321
= 5.7321
Therefore, the exact value of expression 2 is 5.7321.