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Popular Trigonometry >

tan(x)+cot(x)<1

Solution

tan (x) + cot (x) < 1

Solution

\frac{π}{2} + πn < x < π + πn

Interval Notation

(\frac{π}{2} + πn, π + πn)

Decimal

1.57079 \dots + πn < x < 3.14159 \dots + πn

Solution steps

tan (x) + cot (x) < 1

Periodicity of

tan (x) + cot (x) : π

The compound periodicity of the sum of periodic functions is the least common multiplier of the periods

tan (x), cot (x)

Periodicity of

tan (x) : π

Periodicity of

tan (x)

π

= π

Periodicity of

cot (x) : π

Periodicity of

cot (x)

π

= π

Combine periods:

π, π

= π

Express with sin, cos

tan (x) + cot (x) < 1

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} + cot (x) < 1

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

\frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sin ( x )} < 1

\frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sin ( x )} < 1

Simplify

\frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sin ( x )} : \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{sin ( x )}{cos ( x )} + \frac{cos ( x )}{sin ( x )}

Least Common Multiplier of

cos (x), sin (x) : cos (x) sin (x)

cos (x), sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (x)

sin (x)

= cos (x) sin (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

cos (x) sin (x)

For

\frac{sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x ) sin ( x )}{cos ( x ) sin ( x )} = \frac{sin ^{2} ( x )}{cos ( x ) sin ( x )}

For

\frac{cos ( x )}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{cos ( x )}{sin ( x )} = \frac{cos ( x ) cos ( x )}{sin ( x ) cos ( x )} = \frac{cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{sin ^{2} ( x )}{cos ( x ) sin ( x )} + \frac{cos ^{2} ( x )}{cos ( x ) sin ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ( x ) sin ( x )} < 1

Find the zeroes and undifined points of

\frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ( x ) sin ( x )}

for

0 \leq x < π

To find the zeroes, set the inequality to zero

\frac{sin ^{2} ( x ) + cos ^{2} ( x )}{cos ( x ) sin ( x )} = 0

Find the undefined points:

x = \frac{π}{2}, x = 0

Find the zeros of the denominator

cos (x) sin (x) = 0

Solving each part separately

cos (x) = 0 or sin (x) = 0

cos (x) = 0, 0 \leq x < π : x = \frac{π}{2}

cos (x) = 0, 0 \leq x < π

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solutions for the range

0 \leq x < π

x = \frac{π}{2}

sin (x) = 0, 0 \leq x < π : x = 0

sin (x) = 0, 0 \leq x < π

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Solutions for the range

0 \leq x < π

x = 0

Combine all the solutions

x = \frac{π}{2}, x = 0

0, \frac{π}{2}

Identify the intervals

0 < x < \frac{π}{2}, \frac{π}{2} < x < π

Summarize in a table:

sin^{2} (x) + cos^{2} (x) cos (x) sin (x) \frac{s i n ^{2} ( x ) + c o s ^{2} ( x )}{c o s ( x ) s i n ( x )} x = 0 + + 0 Undefined 0 < x < \frac{π}{2} + + + + x = \frac{π}{2} + 0 + Undefined \frac{π}{2} < x < π + - + - x = π + - 0 Undefined

Identify the intervals that satisfy the required condition:

< 0

\frac{π}{2} < x < π

Apply the periodicity of

tan (x) + cot (x)

\frac{π}{2} + πn < x < π + πn

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