Question

Find the Exact value of each equation between
`0\leq theta\leq2\pi`

15)
`cos(-(13\pi )/(3) )`

16)
`csc((23\pi )/(4)`)

17)
`sec-((7\pi )/(2)`)

18)
`cot(-(29\pi )/(6)`)

Answer 1

Use the fact that the co/sine functions are
`2\pi`-periodic and that the tangent function is
`\pi`-periodic. Also, recall that
`\cos x` is even (so that
`\cos(-x)=\cos x`) and
`\sin x` is odd (so that
`\sin(-x)=-\sin x`.

15.

`\cos\left(-\frac{13\pi}3\right)=\cos\frac{13\pi}3=\cos\left(\frac\pi3+4\pi\right)=\cos\frac\pi3=\boxed{\frac12}`

16.

`\sin\frac{23\pi}4=\sin\left(\frac{3\pi}4+5\pi\right)=\sin\left(\frac{3\pi}4+\pi\right)=\sin\frac{7\pi}4=-\frac1{\sqrt2}`

`\implies\csc\frac{23\pi}4=\boxed{-\sqrt2}`

17.

`\cos\left(-\frac{7\pi}2\right)=\cos\frac{7\pi}2=\cos\left(\frac\pi2+3\pi\right)=\cos\left(\frac\pi2+\pi\right)=\cos\frac{3\pi}2=0`

18.

`\tan\left(-\frac{29\pi}6\right)=\frac{\sin\left(-\frac{29\pi}6\right)}{\cos\left(-\frac{29\pi}6\right)}=-\frac{\sin\frac{29\pi}6}{\cos\frac{29\pi}6}`

`\sin\frac{29\pi}6=\sin\left(\frac{5\pi}6+4\pi\right)=\sin\frac{5\pi}6=-\frac12`

`\cos\frac{29\pi}6=\cos\frac{5\pi}6=\frac{\sqrt3}2`

`\implies\tan\left(-\frac{29\pi}6\right)=-\frac{-\frac12}{\frac{\sqrt3}2}=\frac1{\sqrt3}`

`\implies\cot\left(-\frac{29\pi}6\right)=\boxed{\sqrt3}`