Answer :
Answer:
[tex]\sec \left(\dfrac{5\pi}{8}\right) \approx -2.6131[/tex]
Step-by-step explanation:
Trigonometry
Reciprocal Functions
Reciprocal trig functions like secant (sec), cosecant (csc), and cotangent (cot) are 1 over cosine (cos), sine (sin), and tangent (tan) respectively.
- [tex]\sec(x)=\dfrac{1}{\cos(x)}[/tex]
- [tex]\csc(x)=\dfrac{1}{\sin(x)}[/tex]
- [tex]\cot(x)=\dfrac{1}{\tan(x)}[/tex]
[tex]\dotfill[/tex]
Half Angle Formula
Half angle formulas utilizes the trigonometric function of the original (full) angle.
- [tex]\cos\left(\dfrac{\theta}{2}\right)=\pm \sqrt{\dfrac{1-\cos(\theta) }{2}}[/tex]
- [tex]\cos\left(\dfrac{\theta}{2}\right)=\pm \sqrt{\dfrac{1+\cos(\theta) }{2}}[/tex]
- [tex]\tan\left(\dfrac{\theta}{2}\right)=\pm \sqrt{\dfrac{{1-\cos(\theta)}}{{1+\cos(\theta)}} }[/tex]
The plus-minus sign next to the radical is determined by the location of the half angle.
For example, if the half-angle is located in a quadrant where its cosine value would be positive, the cosine half angle formula would have a positive sign.
[tex]\hrulefill[/tex]
Solving the Problem
We're asked to use the half angle formula to find the value of
[tex]\sec\left(\dfrac{5\pi}{8}\right)[/tex].
Knowing that secant is the reciprocal function of cosine we use adjust the half angle formula of cosine to accommodate.
[tex]\cos\left(\dfrac{\theta}{2}\right)=\pm \sqrt{\dfrac{1+\cos(\theta) }{2}}[/tex]
[tex]\dfrac{1}{\cos\left(\dfrac{\theta}{2}\right)}=\pm\dfrac{1}{\sqrt{\dfrac{1-\cos(\theta) }{2}}}[/tex]
[tex]\sec \left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{2}{{1+\cos(\theta)}}}[/tex]
We want the left side to equal the given secant function, so theta must be equal to double the angle given.
[tex]\sec \left(\dfrac{\dfrac{10\pi}{8}}{2}\right) = \pm \sqrt{\dfrac{2}{{1+\cos\left(\dfrac{10\pi}{8}\right)}}}[/tex]
Since the angle is located in the second quadrant where its cosine value is negative, the radical is negative.
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{2}{{1+\cos\left(\dfrac{10\pi}{8} \right)}}}[/tex]
We can simplify the angle on the right side to evaluate the cosine function more easily.
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{2}{{1+\cos\left(\dfrac{5\pi}{4} \right)}}}[/tex]
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{2}{{1+\left(-\dfrac{\sqrt2}{2} \right)}}[/tex]
We can simplify the denominator of the radical by combining the two terms into one fraction.
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{2}{{\dfrac{2-\sqrt2}{2} }}[/tex]
By using KCF (keep, change, flip) method to evaluate fractions we can simplify the term under the radical.
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{4}{2-\sqrt2}[/tex]
We can rationalize the term to simplify further.
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{4}{2-\sqrt2} \cdot \dfrac{2+\sqrt2}{2+\sqrt2}[/tex]
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{\dfrac{8+4\sqrt2}{2}[/tex]
We can divide each term on the numerator to simplify the term into an expression.
[tex]\sec \left(\dfrac{5\pi}{8}\right) = - \sqrt{4+2\sqrt2}[/tex]
Plugging the right side of the equation into the calculator we can find the decimal equivalent.
[tex]\sec \left(\dfrac{5\pi}{8}\right) \approx -2.6131[/tex]
