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tan^2(x)= 1/3 ,0<= x<= 720

Solution

tan^{2} (x) = \frac{1}{3}, 0^{\circ} \leq x \leq 72 0^{\circ}

x = 3 0^{\circ}, x = 21 0^{\circ}, x = 39 0^{\circ}, x = 57 0^{\circ}, x = 15 0^{\circ}, x = 33 0^{\circ}, x = 51 0^{\circ}, x = 69 0^{\circ}

Radians

x = \frac{π}{6}, x = \frac{7 π}{6}, x = \frac{13 π}{6}, x = \frac{19 π}{6}, x = \frac{5 π}{6}, x = \frac{11 π}{6}, x = \frac{17 π}{6}, x = \frac{23 π}{6}

Solution steps

tan^{2} (x) = \frac{1}{3}, 0^{\circ} \leq x \leq 72 0^{\circ}

Solve by substitution

tan^{2} (x) = \frac{1}{3}

Let:

tan (x) = u

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3} : u = \frac{1}{3}, u = - \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = tan (x)

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3}, tan (x) = - \frac{1}{3}

tan (x) = \frac{1}{3}, 0 \leq x \leq 72 0^{\circ} : x = 3 0^{\circ}, x = 21 0^{\circ}, x = 39 0^{\circ}, x = 57 0^{\circ}

tan (x) = \frac{1}{3}, 0 \leq x \leq 72 0^{\circ}

Apply trig inverse properties

tan (x) = \frac{1}{3}

General solutions for

tan (x) = \frac{1}{3}

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{1}{3}) + 18 0^{\circ} n

x = arctan (\frac{1}{3}) + 18 0^{\circ} n

Solutions for the range

0 \leq x \leq 72 0^{\circ}

x = 3 0^{\circ}, x = 21 0^{\circ}, x = 39 0^{\circ}, x = 57 0^{\circ}

tan (x) = - \frac{1}{3}, 0 \leq x \leq 72 0^{\circ} : x = 15 0^{\circ}, x = 33 0^{\circ}, x = 51 0^{\circ}, x = 69 0^{\circ}

tan (x) = - \frac{1}{3}, 0 \leq x \leq 72 0^{\circ}

Apply trig inverse properties

tan (x) = - \frac{1}{3}

General solutions for

tan (x) = - \frac{1}{3}

tan (x) = - a \Rightarrow x = arctan (- a) + 18 0^{\circ} n

x = arctan (- \frac{1}{3}) + 18 0^{\circ} n

x = arctan (- \frac{1}{3}) + 18 0^{\circ} n

Solutions for the range

0 \leq x \leq 72 0^{\circ}

x = 15 0^{\circ}, x = 33 0^{\circ}, x = 51 0^{\circ}, x = 69 0^{\circ}

Combine all the solutions

x = 3 0^{\circ}, x = 21 0^{\circ}, x = 39 0^{\circ}, x = 57 0^{\circ}, x = 15 0^{\circ}, x = 33 0^{\circ}, x = 51 0^{\circ}, x = 69 0^{\circ}

sin (2 x) = 1, 0 \leq x \leq 2 π

cos^{2} (z) = - 1

cosh (x) = \frac{6}{2}

tan (x) - sec (x) = 3, 0 \leq x < 2 π

sin^{2} (θ) + 2 cos^{2} (θ) = 4