Question

evaluate the trig functions below without using technology. enter an exact answer, not a decimal approximation. enter \dne\ if the function is undefined at the given angle.
\\(\sin\left(-\frac{13\pi}{6}\
ight)=\\)
\\(\cos\left(-\frac{10\pi}{3}\
ight)=\\)
\\(\tan\left(\frac{7\pi}{6}\
ight)=\\)
\\(\csc\left(-\frac{23\pi}{6}\
ight)=\\)
\\(\sec\left(\frac{5\pi}{4}\
ight)=\\)
\\(\cot\left(\frac{17\pi}{6}\
ight)=\\)
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Explanation:

Step1: Rewrite -13π/6

Use the property of sine function and angle -13π/6 = -2π - π/6. Since sin(x - 2π)=sin(x), then sin(-13π/6)=sin(-π/6). And sin(-x)= - sin(x), so sin(-π/6)=-sin(π/6)= - 1/2.

Step2: Rewrite -10π/3

-10π/3=-4π + 2π/3. Since cos(x - 4π)=cos(x), then cos(-10π/3)=cos(2π/3). And cos(2π/3)=-1/2.

Step3: Rewrite 7π/6

7π/6 = π+π/6. Using the formula tan(A + B)=(tan A+tan B)/(1 - tan A tan B) with A = π and B=π/6, and tan(π)=0, tan(7π/6)=tan(π + π/6)=tan(π/6)=√3/3.

Step4: Rewrite -23π/6

-23π/6=-4π+π/6. Since csc(x - 4π)=csc(x), then csc(-23π/6)=csc(π/6). And csc(π/6) = 1/sin(π/6)=2.

Step5: Rewrite 5π/4

5π/4=π + π/4. Since sec(x)=1/cos(x), and cos(π + π/4)=-cos(π/4)=-√2/2, then sec(5π/4)=1/cos(5π/4)=-√2.

Step6: Rewrite 17π/6

17π/6 = 3π - π/6. Since cot(x)=cos(x)/sin(x), and cos(3π - π/6)=-cos(π/6), sin(3π - π/6)=sin(π/6), then cot(17π/6)=cos(17π/6)/sin(17π/6)=-√3.

Answer:

sin(-13π/6)=-1/2
cos(-10π/3)=-1/2
tan(7π/6)=√3/3
csc(-23π/6)=2
sec(5π/4)=-√2
cot(17π/6)=-√3