# In A and B, cosine is given..find sine and tangent if x liesin the specific interval. A.) cos (x) = 1/3 x in [ pi/2, 0] B.) cos (x) = -5/13 x in [ pi/2, pi] C.) s = sec( pi x/2 )...Find the period. Whatsymmetries does the function have? D.) cos (x + pi /4) -1. What is the period?

Violet Woodward 2022-07-28 Answered
In A and B, cosine is given..find sine and tangent if x liesin the specific interval.
A.) $\mathrm{cos}\left(x\right)=1/3x\in \left[\pi /2,0\right]$
B.) $\mathrm{cos}\left(x\right)=-5/13x\in \left[\pi /2,\pi \right]$
C.) $s=\mathrm{sec}\left(\pi x/2\right)$...Find the period. What symmetries does the function have?
D.) $\mathrm{cos}\left(x+\pi /4\right)-1$. What is the period?
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## Answers (1)

eyiliweyouc
Answered 2022-07-29 Author has 15 answers
(A) since $\left[0,\pi /2\right]$ comes in first interval so sine and tan are positive
so;$\mathrm{sin}\left(x\right)=\sqrt{1-{\mathrm{cos}}^{2}\left(x\right)}=\left(\sqrt{9-{x}^{2}}\right)/3$
so;$\mathrm{tan}\left(x\right)=\mathrm{sin}\left(x\right)/\mathrm{cos}\left(x\right)=\left(\sqrt{9-{x}^{2}}\right)/x$
(B) since$\left[\pi /2,\pi \right]$ comes in second interval sosine is positive and tan is negative.
so; $\mathrm{sin}\left(x\right)=\sqrt{1-{\mathrm{cos}}^{2}\left(x\right)}=\left(\sqrt{169-25{x}^{2}}\right)/13$
so;$\mathrm{tan}\left(x\right)=\mathrm{sin}\left(x\right)/\mathrm{cos}\left(x\right)=\left(\sqrt{169-25{x}^{2}}\right)/-5x$
(c)the period of $\mathrm{sec}\left(\pi x/2\right)=2$
(d)the period of $\mathrm{tan}\left(x+\pi /4\right)=\pi /2$
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