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sec(x)sin(x)-3sin(x)=0,0<= x<= 360

Solution

sec (x) sin (x) - 3 sin (x) = 0, 0 \leq x \leq 36 0^{\circ}

x = 0, x = 18 0^{\circ}, x = 36 0^{\circ}, x = 1.23095 \dots, x = 36 0^{\circ} - 1.23095 \dots

Radians

x = 0, x = π, x = 2 π, x = 1.23095 \dots, x = 2 π - 1.23095 \dots

Solution steps

sec (x) sin (x) - 3 sin (x) = 0, 0 \leq x \leq 36 0^{\circ}

Factor

sec (x) sin (x) - 3 sin (x) : sin (x) (sec (x) - 3)

sec (x) sin (x) - 3 sin (x)

Factor out common term

sin (x)

= sin (x) (sec (x) - 3)

sin (x) (sec (x) - 3) = 0

Solving each part separately

sin (x) = 0 or sec (x) - 3 = 0

sin (x) = 0, 0 \leq x \leq 36 0^{\circ} : x = 0, x = 18 0^{\circ}, x = 36 0^{\circ}

sin (x) = 0, 0 \leq x \leq 36 0^{\circ}

General solutions for

sin (x) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

x = 0 + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solve

x = 0 + 36 0^{\circ} n : x = 36 0^{\circ} n

x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

x = 36 0^{\circ} n

x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solutions for the range

0 \leq x \leq 36 0^{\circ}

x = 0, x = 18 0^{\circ}, x = 36 0^{\circ}

sec (x) - 3 = 0, 0 \leq x \leq 36 0^{\circ} : x = arcsec (3), x = 36 0^{\circ} - arcsec (3)

sec (x) - 3 = 0, 0 \leq x \leq 36 0^{\circ}

Move

3

to the right side

sec (x) - 3 = 0

Add

3

to both sides

sec (x) - 3 + 3 = 0 + 3

Simplify

sec (x) = 3

sec (x) = 3

Apply trig inverse properties

sec (x) = 3

General solutions for

sec (x) = 3

sec (x) = a \Rightarrow x = arcsec (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arcsec (a) + 36 0^{\circ} n

x = arcsec (3) + 36 0^{\circ} n, x = 36 0^{\circ} - arcsec (3) + 36 0^{\circ} n

x = arcsec (3) + 36 0^{\circ} n, x = 36 0^{\circ} - arcsec (3) + 36 0^{\circ} n

Solutions for the range

0 \leq x \leq 36 0^{\circ}

x = arcsec (3), x = 36 0^{\circ} - arcsec (3)

Combine all the solutions

x = 0, x = 18 0^{\circ}, x = 36 0^{\circ}, x = arcsec (3), x = 36 0^{\circ} - arcsec (3)

Show solutions in decimal form

x = 0, x = 18 0^{\circ}, x = 36 0^{\circ}, x = 1.23095 \dots, x = 36 0^{\circ} - 1.23095 \dots

cos (u) = \frac{5}{13}

4 sin (\frac{x}{2}) - 1 = 1

- sin (θ) = \frac{1}{2}

3 sec (θ) + 6 = 0

sin (x) - sin (x) \cdot cos (x) = 0, 0 \leq x \leq 2 π