Find the exact value. Without using a calculator
a) cos (-17pi/6)
b) sec (3pi/2)
17/6 = 2 5/6
that is twice around counterclockwise then -5pi/6
that is pi/6 below -x axis
and pi/6 = 180/6 = 30 deg
so 30 deg below -x axis
cos of that is -sqrt3/2
3 pi/2 is 180 + 90 = 270 or on -y axis
undefined
a) To find the exact value of cos(-17π/6), we can use the properties of the cosine function.
First, let's find an angle in the same quadrant as (-17π/6). One full revolution around the unit circle is equal to 2π, so we can find an equivalent angle by adding or subtracting 2π until we get an angle in the range of 0 to 2π.
Adding 2π multiple times to (-17π/6):
(-17π/6) + 2π = (11π/6) + 2π
= (11π + 12π)/6
= 23π/6
Now we have an equivalent angle in the range of 0 to 2π.
Cosine is positive in the first quadrant and negative in the second quadrant. Since 23π/6 is in the second quadrant, the cosine function will be negative.
Now we can find the cosine value using the reference angle in the first quadrant.
The reference angle for 23π/6 is (23π/6) - 2π = (23π - 12π)/6 = 11π/6
The cosine of 11π/6 is equal to the cosine of the reference angle, which is positive.
Therefore, cos(-17π/6) = -cos (11π/6).
b) To find the exact value of sec(3π/2), we can use the fact that sec(θ) is the reciprocal of cos(θ).
First, let's find an angle in the same quadrant as (3π/2). One full revolution around the unit circle is equal to 2π, so we can find an equivalent angle by adding or subtracting 2π until we get an angle in the range of 0 to 2π.
Adding 2π multiple times to (3π/2):
(3π/2) + 2π = (3π + 4π)/2
= 7π/2
Now we have an equivalent angle in the range of 0 to 2π.
The cosine of 7π/2 is undefined since it's the angle where cosine is equal to zero.
Since sec(θ) is the reciprocal of cos(θ), the secant of 7π/2 is also undefined.
To find the exact value without using a calculator, we can use the unit circle and the basic trigonometric ratios.
a) cos(-17π/6):
We need to find the cosine value of -17π/6. To do this, let's convert -17π/6 to a reference angle within the unit circle.
Step 1: Determine the equivalent angle within one revolution (2π):
-17π/6 = -2π + (π/6)
Step 2: Find the reference angle:
The reference angle is the positive angle between the terminal side and the x-axis. In this case, it is π/6.
Step 3: Determine the quadrant:
Since the reference angle is positive, the terminal side of the angle lies in either the first or second quadrant.
Step 4: Determine the sign:
In the first quadrant, both cosine and secant are positive, but in the second quadrant, cosine is negative while secant is positive.
Since -17π/6 is in the second quadrant, the cosine of -17π/6 will be negative.
Step 5: Evaluate the cosine value:
The cosine of -17π/6 is equal to the cosine of the reference angle (π/6) but with a negative sign:
cos(-17π/6) = -cos(π/6)
To find the exact value of cos(π/6), we can use the special triangle with angles of π/6, π/3, and π/2. In that triangle, the cosine of π/6 is √3/2.
Therefore, cos(-17π/6) = -√3/2.
b) sec(3π/2):
We need to find the secant value of 3π/2. Again, let's determine the reference angle within the unit circle.
Step 1: Determine the equivalent angle within one revolution (2π):
3π/2 = π + π/2
Step 2: Find the reference angle:
The reference angle is the positive angle between the terminal side and the x-axis. In this case, it is π/2.
Step 3: Determine the quadrant:
Since the reference angle is positive, the terminal side of the angle lies in either the first or fourth quadrant.
Step 4: Determine the sign:
In the first quadrant, both cosine and secant are positive, while in the fourth quadrant, cosine is negative while secant is positive.
Since 3π/2 is in the fourth quadrant, the secant of 3π/2 will be positive.
Step 5: Evaluate the secant value:
The secant of 3π/2 is equal to 1 divided by the cosine of the reference angle (π/2):
sec(3π/2) = 1/cos(π/2)
Using the special triangle again, with an angle of π/2, the cosine value is 0 (since it is located on the y-axis). Therefore, the secant of π/2 is undefined.
Hence, sec(3π/2) is undefined.