evaluate exactly(without calculator):
1. sec(pi/4)
2.3sin4pi
3. tan240(degrees)
1. Cos(pi/4)= cos 45 degrees = 1/sqrt2
What does that tell you about sec(pi/4)?
2. sin 4 pi = sin 0 - ___
3. tan 240 deg = tan 60 deg = ___
cos pi/4 = cos 45 degrees = sqrt 2/2
so
sec = 1/cos = 2/sqrt 2 = 2 sqrt 2
sin 4 pi = sin 0 (twice all the way around ) = 0
240 = 180 + 60
in quadrant 3, x is - , y is -
so tangent is +
tan 60 = (sqrt 3)/1 = sqrt 3
To evaluate these expressions without a calculator, we can use the trigonometric identities and special angles to simplify them. Let's go through each expression one by one:
1. Evaluating sec(pi/4):
The secant function is the reciprocal of the cosine function: sec(x) = 1/cos(x). We know that cos(pi/4) = sqrt(2)/2 from the special angles. Therefore, sec(pi/4) = 1/(sqrt(2)/2) = 2/sqrt(2).
To rationalize the denominator, multiply the numerator and denominator by sqrt(2): (2/sqrt(2)) * (sqrt(2)/sqrt(2)) = 2sqrt(2)/2 = sqrt(2).
So, sec(pi/4) = sqrt(2).
2. Evaluating 3sin(4pi):
Since sin(4pi) = 0, 3sin(4pi) = 3 * 0 = 0.
So, 3sin(4pi) = 0.
3. Evaluating tan(240 degrees):
To evaluate this expression, we need to convert 240 degrees to radians. Since pi radians = 180 degrees, we can use the conversion factor (pi/180) to convert degrees to radians.
240 degrees * (pi/180) = (4/3)pi radians.
The tangent function is the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). In this case, we have tan((4/3)pi).
The sine of (4/3)pi = sqrt(3)/2 and the cosine of (4/3)pi = -1/2. So, tan((4/3)pi) = (sqrt(3)/2) / (-1/2).
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction: (sqrt(3)/2) * (-2/1) = -sqrt(3).
Therefore, tan(240 degrees) = -sqrt(3).