What is the general solution for 3+4cot(x)=5csc(x) ?

The general solution for 3+4cot(x)=5csc(x) is x=0.64350…+2pin

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3+4cot(x)=5csc(x)

3 + 4 cot (x) = 5 csc (x)

x = 0.64350 \dots + 2 πn

Solution steps

3 + 4 cot (x) = 5 csc (x)

Subtract

5 csc (x)

from both sides

3 + 4 cot (x) - 5 csc (x) = 0

Express with sin, cos

3 + 4 \cdot \frac{cos ( x )}{sin ( x )} - 5 \cdot \frac{1}{sin ( x )} = 0

Simplify

3 + 4 \cdot \frac{cos ( x )}{sin ( x )} - 5 \cdot \frac{1}{sin ( x )} : \frac{3 sin ( x ) + 4 cos ( x ) - 5}{sin ( x )}

\frac{3 sin ( x ) + 4 cos ( x ) - 5}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 sin (x) + 4 cos (x) - 5 = 0

Subtract

4 cos (x)

from both sides

3 sin (x) - 5 = - 4 cos (x)

Square both sides

(3 sin (x) - 5)^{2} = (- 4 cos (x))^{2}

Subtract

(- 4 cos (x))^{2}

from both sides

(3 sin (x) - 5)^{2} - 16 cos^{2} (x) = 0

Rewrite using trig identities

9 + 25 sin^{2} (x) - 30 sin (x) = 0

Solve by substitution

sin (x) = \frac{3}{5}

sin (x) = \frac{3}{5} : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (\frac{3}{5}) + 2 πn

Show solutions in decimal form

x = 0.64350 \dots + 2 πn

(tan (x) - 1) (sec (x) + 1) = 0

tan (θ) = \frac{9}{10}

tan (x) = \frac{3}{205}

sin (x) sin (2 x) = sin (3 x)

cot (x) = \frac{15}{8}

What is the general solution for 3+4cot(x)=5csc(x) ?
The general solution for 3+4cot(x)=5csc(x) is x=0.64350…+2pin