Can some one help me wit this:
Find the exact value of 4cos7(pi)/4 - 2sin(pi)/3
please and thank you
To find the exact value of the given expression, we'll use the unit circle and the trigonometric identities. Let's break down the expression step by step.
First, let's simplify 4cos(7π/4):
To find the cosine of an angle, we refer to the unit circle. The angle 7π/4 is in the third quadrant, where cosine is negative. In the third quadrant, the reference angle is π/4 (since it's the angle formed between the terminal side and the x-axis). The value of cosine for π/4 is √2/2.
Since cosine is negative in the third quadrant, the value of 4cos(7π/4) is -4√2/2, which simplifies to -2√2.
Now, let's simplify -2sin(π/3):
To find the sine of an angle, we again refer to the unit circle. The angle π/3 is in the first quadrant, where sine is positive. In the first quadrant, the reference angle is π/3 (same as the given angle). The value of sine for π/3 is √3/2.
Since sine is positive in the first quadrant, the value of -2sin(π/3) is -2*(√3/2), which simplifies to -√3.
Finally, substituting these values back into the original expression:
4cos(7π/4) - 2sin(π/3) = -2√2 - (-√3).
Simplifying further, we can write this as:
-2√2 + √3.
So, the exact value of the given expression is -2√2 + √3.