Can some one help me wit this:
Find the exact value of 4cos7(3.14...)/4 - 2sin(3.14...)/3
please and thank you
Sure! To find the exact value of the given expression, we can start by simplifying each term individually.
Let's start with the first term: 4cos(7π/4). To find the cosine of any angle, we can use the unit circle or a trigonometric identity. In this case, we'll use the unit circle.
The angle 7π/4 falls in the third quadrant of the unit circle. In the third quadrant, the cosine values are negative. So, cos(7π/4) is equal to -√2/2.
Now, let's simplify the second term: -2sin(3π/4)/3. Like before, we'll use the unit circle to find the sine of 3π/4.
The angle 3π/4 falls in the second quadrant of the unit circle. In the second quadrant, the sine values are positive. Thus, the sine of 3π/4 is √2/2.
Now that we have the simplified forms of each term, we can substitute them back into the expression:
(4cos(7π/4))/4 - (2sin(3π/4))/3
Substituting the values:
(4*(-√2/2))/4 - (2*(√2/2))/3
Now, let's simplify further:
-√2/2 - √2/3
To combine the terms, we need a common denominator, which in this case is 6. Multiplying the first fraction by 3/3 and the second fraction by 2/2, we get:
(-3√2/6) - (2√2/6)
Combining the fractions:
(-3√2 - 2√2)/6
Now, we can simplify the numerator:
(-5√2)/6
So, the exact value of the expression 4cos(7π/4)/4 - 2sin(3π/4)/3 is (-5√2)/6.