Use one of the identities
cos(t + 2πk) = cos t or sin(t + 2πk) = sin t
to evaluate each expression. (Enter your answers in exact form.)
(a)
sin(19π/4)
(b)
sin(−19π/4)
(c)
cos(11π)
(d)
cos(53π/4)
(e)
tan(−3π/4)
(f)
cos(π/4)
(g)
sec(π/6+ 2π)
(h)
csc(2π − 2π/3)
here is how I do these ....
(perhaps you might find it easier in degrees, 19π/4 radians = 855°
sin(19π/4) = sin 855°
take away 2 rotations, (720° or 4π)
sin 19π/4 = sin 3π/4 = sin 135°
3π/4 or 135° is in quadrant II and π/4 or 45° from the x-axis
We also know that in II, the sine is positive
so sin 3π/4 = sin π/4 = sin 45° = 1/√2 or √2/2
.....
d) cos(53π/4) or cos 2385°
take away 12π or 2160°
cos 53π/4 = cos 5π/4 or cos 225°
This angle is in quad III making it negative and 45 from the x-axis
cos 53π/4 = - cos π/4 = -1/√2 or -√2/2
do the others the same way.
cos11pi/6
To evaluate each expression using the given identities, we can begin by simplifying the angles.
(a) sin(19π/4):
Using the identity sin(t + 2πk) = sin(t), we can simplify the angle to 19π/4 = 4π + 3π/4.
Since sin(t) repeats after every 2π, we can remove the 4π and the angle becomes 3π/4.
So, sin(19π/4) = sin(3π/4).
(b) sin(-19π/4):
Using the identity sin(t + 2πk) = sin(t), we can simplify the angle to -19π/4 = -4π - 3π/4.
Since sin(t) repeats after every 2π, we can remove the -4π and the angle becomes -3π/4.
So, sin(-19π/4) = sin(-3π/4).
(c) cos(11π):
Using the identity cos(t + 2πk) = cos(t), we can simplify the angle to 11π = 5π + π.
Since cos(t) repeats after every 2π, we can remove the 5π and the angle becomes π.
So, cos(11π) = cos(π).
(d) cos(53π/4):
Using the identity cos(t + 2πk) = cos(t), we can simplify the angle to 53π/4 = 13π + π/4.
Since cos(t) repeats after every 2π, we can remove the 13π and the angle becomes π/4.
So, cos(53π/4) = cos(π/4).
(e) tan(-3π/4):
Using the identity sin(t + 2πk) = sin(t) and cos(t + 2πk) = cos(t), we can simplify the angle.
-3π/4 lies in the third quadrant where both sin(t) and cos(t) are negative.
So, tan(-3π/4) = sin(-3π/4) / cos(-3π/4).
(f) cos(π/4):
No simplification is needed here, so we can directly evaluate cos(π/4).
(g) sec(π/6 + 2π):
Using the identity sec(t + 2πk) = sec(t), we can simplify the angle to π/6 + 2π = 13π/6.
So, sec(π/6 + 2π) = sec(13π/6).
(h) csc(2π - 2π/3):
Using the identity sin(t + 2πk) = sin(t), we can simplify the angle to 2π - 2π/3 = 6π/3 - 2π/3 = 4π/3.
So, csc(2π - 2π/3) = csc(4π/3).
To evaluate each expression using the given identities, we need to find an equivalent angle within the interval [0, 2π] that we can use with the cosine and sine functions. Let's go through each expression:
(a) sin(19π/4)
We can rewrite 19π/4 as 4π + 3π/4. Notice that 4π is equivalent to 2 complete revolutions (2πk) around the unit circle and does not affect the sine function. Therefore, we can use the identity sin(t + 2πk) = sin(t) to rewrite sin(19π/4) as sin(3π/4).
(b) sin(-19π/4)
Similarly, we can rewrite -19π/4 as -4π - 3π/4. Again, -4π is equivalent to -2 complete revolutions around the unit circle, so we can simplify sin(-19π/4) to sin(-3π/4).
(c) cos(11π)
Since 11π is equivalent to 5 complete revolutions (2πk + π) around the unit circle, we can use the identity cos(t + 2πk) = cos(t) to rewrite cos(11π) as cos(π).
(d) cos(53π/4)
We can rewrite 53π/4 as 12π + π/4. The 12π part represents 6 complete revolutions around the unit circle, so cos(53π/4) is equivalent to cos(π/4).
(e) tan(-3π/4)
Using the identity tan(t) = sin(t)/cos(t), we can rewrite tan(-3π/4) as sin(-3π/4) / cos(-3π/4). Applying the identity sin(t + 2πk) = sin(t), we can simplify it to sin(π/4) / cos(π/4).
(f) cos(π/4)
No simplification needed for this expression.
(g) sec(π/6 + 2π)
Using the identity sec(t) = 1/cos(t), we can rewrite sec(π/6 + 2π) as 1 / cos(π/6 + 2π).
(h) csc(2π - 2π/3)
Using the identity csc(t) = 1/sin(t), we can rewrite csc(2π - 2π/3) as 1 / sin(2π - 2π/3).
Now, we can evaluate each expression:
(a) sin(19π/4) = sin(3π/4)
(b) sin(-19π/4) = sin(-3π/4)
(c) cos(11π) = cos(π)
(d) cos(53π/4) = cos(π/4)
(e) tan(-3π/4) = sin(π/4) / cos(π/4)
(f) cos(π/4)
(g) sec(π/6 + 2π) = 1 / cos(π/6 + 2π)
(h) csc(2π - 2π/3) = 1 / sin(2π - 2π/3)
Note: To further simplify the expressions, you can use the familiar values and properties of trigonometric functions (e.g., sin(π/4) = √2/2, cos(π) = -1).