Evaluate 8(1+sin 8/3 radians)without using maths table or a calculator
To evaluate 8(1+sin(8/3) radians) without using a calculator or math table, we can follow these steps:
Step 1: Convert the angle from radians to degrees
To convert radians to degrees, we use the conversion factor: 180 degrees / π radians.
Therefore, the angle becomes (8/3) * (180 degrees / π radians) = (8 * 180) / (3 * π) degrees.
Step 2: Simplify the expression inside the sine function
Since sin(0) = 0, we can simplify sin[(8 * 180) / (3 * π)] as sin[(8 * 60) / (3 * π)].
Step 3: Simplify the expression with the sine function
Using the periodicity property of the sine function, we can rewrite sin[(8 * 60) / (3 * π)] as sin[(2 * 60) / (3 * π)].
Step 4: Evaluate the simplified expression
With the angle in the form (2 * θ) / π, we can use the special values of the sine function for angles 0°, 30°, 45°, 60°, 90°, and their negatives to evaluate the expression. Since we have (2 * 60) / (3 * π), which is equivalent to (π / 3) radians:
sin(π / 3) = √3 / 2
Step 5: Evaluate the final expression
Now, we have 8(1 + sin(8/3 radians)) = 8(1 + √3 / 2).
To simplify this expression, distribute the 8:
8 + 8(√3 / 2) = 8 + 4√3.
So, the evaluation of 8(1 + sin(8/3 radians)) without using a math table or calculator is 8 + 4√3.
To solve this expression without using a calculator or math table, we can use some trigonometric identities and simplifications.
First, notice that the expression contains the term "sin(8/3 radians)". We can simplify this using the following identity:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
In this case, we have x = 8/3 radians and y = 0 radians. Therefore, we can rewrite sin(8/3 radians) as:
sin(8/3 radians) = sin(8/3)sin(0) + cos(8/3)sin(0)
Since sin(0) = 0, this simplifies to:
sin(8/3 radians) = 0
Now, we can substitute this result into the original expression:
8(1 + sin(8/3 radians)) = 8(1 + 0) = 8(1) = 8
Therefore, the value of 8(1 + sin(8/3 radians)) is 8.