Find the exact value of the trigonometric function:
sec(-pi/12)
I am not sure how to answer this off the unit circle.
You should know that
sec (-pi/12)
= 1/cos(-pi/12)
= 1/cos(pi/12)
You might find it "easier" to visualize degrees than radians, if so, the
pi/12 radians = 15º
so we need cos 15º
cos15 = cos(45-30) ... (I am dropping the º symbol for easier typing)
= cos45cos30 + sin45sin30
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4
and
cos(-pi/12) = 4/(√6 + √2)
To find the exact value of the trigonometric function sec(-π/12), we need to determine the value of sec(-π/12) on the unit circle.
First, let's understand the term sec. Secant (sec) is the reciprocal of cosine (cos). It can be defined as sec(x) = 1 / cos(x).
Now, let's find the value of cos(-π/12) on the unit circle.
-π/12 is an angle in the fourth quadrant, measured clockwise from the positive x-axis. To determine the reference angle, we need to find the angle formed by the terminal side of -π/12 and the x-axis.
Since -π/12 is in the fourth quadrant, its reference angle will be positive π/12, which is the distance between -π/12 and the x-axis.
The reference angle, π/12, corresponds to an angle of 15 degrees.
On the unit circle, the cosine value (x-coordinate) at 15 degrees is (√3 - 1) / 2√2.
Since sec(x) = 1 / cos(x), we can find sec(-π/12) by taking the reciprocal of cos(-π/12):
sec(-π/12) = 1 / ((√3 - 1) / 2√2)
To simplify this expression, we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
sec(-π/12) = (2√2) / (√3 - 1) * (√3 + 1) / (√3 + 1)
Simplifying the numerator:
sec(-π/12) = [2√2 * (√3 + 1)] / [(√3 - 1) * (√3 + 1)]
Finally, let's simplify the denominator and rationalize the numerator:
sec(-π/12) = [2√6 + 2√2] / [(√3)^2 - (1)^2]
sec(-π/12) = [2√6 + 2√2] / [3 - 1]
sec(-π/12) = [2√6 + 2√2] / 2
simplifying further:
sec(-π/12) = √6 + √2
Therefore, the exact value of sec(-π/12) is √6 + √2.
To find the exact value of the trigonometric function sec(-π/12), we need to use the properties of trigonometric functions.
Recall that sec(x) is the reciprocal of cos(x), so to find sec(-π/12), we need to find the value of cos(-π/12) first.
The cosine function has a periodicity of 2π, which means that cos(x) = cos(x + 2π). Therefore, we can rewrite cos(-π/12) as cos(-π/12 + 2π).
Now let's simplify this expression. Since -π/12 is negative, adding 2π will bring it into the positive range. We can rewrite it as:
cos(-π/12 + 2π) = cos(23π/12)
To find the exact value of cos(23π/12), we can use the sum-to-product formula for cosines:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
In this case, let A = 15π/12 and B = 8π/12. Now we have:
cos(23π/12) = cos(15π/12 + 8π/12)
Using the sum-to-product formula, we get:
cos(23π/12) = cos(15π/12)cos(8π/12) - sin(15π/12)sin(8π/12)
cos(23π/12) = cos(5π/4)cos(2π/3) - sin(5π/4)sin(2π/3)
Recall that cos(5π/4) = -√2/2 and sin(5π/4) = -√2/2.
Also, cos(2π/3) = 1/2 and sin(2π/3) = √3/2.
Substituting these values, we have:
cos(23π/12) = (-√2/2)(1/2) - (-√2/2)(√3/2)
Simplifying further:
cos(23π/12) = -√2/4 - √6/4
Finally, since sec(x) is the reciprocal of cos(x), the exact value of sec(-π/12) is:
sec(-π/12) = 1/cos(-π/12) = 1/[-√2/4 - √6/4]