Solve the trigonometric equations on the interval 0 ≤ θ &lt; 2 π.(32) 4 ( 1 + sin ⁡ θ ) = cos 2 ⁡ θ(38) sec ⁡ θ = tan ⁡ θ + cot ⁡ θ(40) 3 sin ⁡ θ + cos ⁡ θ = 1

Question

Solve the trigonometric equations on the interval   0  ≤  θ  &amp;lt;  2  π.(32)  4  (  1  +  sin  ⁡  θ  )  =      cos          2        ⁡  θ(38)  sec  ⁡  θ  =  tan  ⁡  θ  +  cot  ⁡  θ(40)      3    sin  ⁡  θ  +  cos  ⁡  θ  =  1

Osvaldo Crosby · Accepted Answer

(32) Use       sin          2        +      cos          2        =  1 to write as  4  (  1  +  sin  ⁡  θ  )  =  1  −      sin          2        ⁡  θThis is a quadratic equation in   sin  ⁡  θ  :      sin          2        ⁡  θ  +  4  sin  ⁡  θ  +  3  =  0  (  sin  ⁡  θ  +  2      )          2        =  1  sin  ⁡  θ  =  −  1  →  θ  =  3  π      /    2(38) Multiply throughout by   cos  ⁡  θ  :  1  =  sin  ⁡  θ  +      cos          2        ⁡  θ      /    sin  ⁡  θMultiply throughout by   sin  ⁡  θ  :  sin  ⁡  θ  =      sin          2        ⁡  θ  +      cos          2        ⁡  θRewrite using       sin          2        +      cos          2        =  1  sin  ⁡  θ  =  1  →  θ  =  π      /    2(40) Divide throughout by   cos  ⁡  θ  :      3    tan  ⁡  θ  +  1  =  sec  ⁡  θSquare both sides:  3      tan          2        ⁡  θ  +  2      3    tan  ⁡  θ  +  1  =      sec          2        ⁡  θUse       sec          2        =  1  +      tan          2        :  2      tan          2        ⁡  θ  +  2      3    tan  ⁡  θ  =  0  tan  ⁡  θ  =  −      3    →  θ  =  −  π      /    3  +  n  π so here   θ  =  2  π      /    3

Parker Bird · Accepted Answer

32.   4  (  1  +  sin  ⁡  θ  )  =  1  −      sin          2        ⁡  θso,       sin          2        ⁡  θ  +  4  sin  ⁡  θ  +  3  =  0so,   (  sin  ⁡  θ  +  1  )  (  sin  ⁡  θ  +  3  )  =  0so,   sin  ⁡  θ  =  −  1so,   θ  =  3  π      /    238.   1      /    cos  ⁡  θ  =  sin  ⁡  θ      /    cos  ⁡  θ  +  cos  ⁡  θ      /    sin  ⁡  θso,   sin  ⁡  θ  =      sin          2        ⁡  θ  +      cos          2        ⁡  θ  =  1so,   θ  =  π      /    240.       3    sin  ⁡  θ  +  cos  ⁡  θ  =  1now, divide both sides by 2so,   sin  ⁡  θ  cos  ⁡  (  π      /    6  )  +  sin  ⁡  (  π      /    6  )  cos  ⁡  θ  =  sin  ⁡  (  π      /    6  )so,   sin  ⁡  (  θ  +  π      /    6  )  =  sin  ⁡  (  π      /    6  )so,   θ  =  0  ,  π  −  π      /    6  ,  i  e  .  0  ,  5  π      /    6

Solve the trigonometric equations on the interval 0 le theta < 2pi (32) 4(1 + sin theta) = cos^2 theta (38) sec theta = tan theta + cot theta (40)sqrt(3)sin theta + cos theta = 1

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