What is the general solution for 3cos(x)cot(x)+7=5csc(x) ?

The general solution for 3cos(x)cot(x)+7=5csc(x) is x=0.33983…+2pin,x=pi-0.33983…+2pin

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3cos(x)cot(x)+7=5csc(x)

3 cos (x) cot (x) + 7 = 5 csc (x)

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

Solution steps

3 cos (x) cot (x) + 7 = 5 csc (x)

Subtract

5 csc (x)

from both sides

3 cos (x) cot (x) + 7 - 5 csc (x) = 0

Express with sin, cos

3 cos (x) \frac{cos ( x )}{sin ( x )} + 7 - 5 \cdot \frac{1}{sin ( x )} = 0

Simplify

3 cos (x) \frac{cos ( x )}{sin ( x )} + 7 - 5 \cdot \frac{1}{sin ( x )} : \frac{3 cos ^{2} ( x ) - 5 + 7 sin ( x )}{sin ( x )}

\frac{3 cos ^{2} ( x ) - 5 + 7 sin ( x )}{sin ( x )} = 0

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

3 cos^{2} (x) - 5 + 7 sin (x) = 0

Subtract

7 sin (x)

from both sides

3 cos^{2} (x) - 5 = - 7 sin (x)

Square both sides

(3 cos^{2} (x) - 5)^{2} = (- 7 sin (x))^{2}

Subtract

(- 7 sin (x))^{2}

from both sides

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x) = 0

Factor

(3 cos^{2} (x) - 5)^{2} - 49 sin^{2} (x) : (3 cos^{2} (x) - 5 + 7 sin (x)) (3 cos^{2} (x) - 5 - 7 sin (x))

(3 cos^{2} (x) - 5 + 7 sin (x)) (3 cos^{2} (x) - 5 - 7 sin (x)) = 0

Solving each part separately

3 cos^{2} (x) - 5 + 7 sin (x) = 0 or 3 cos^{2} (x) - 5 - 7 sin (x) = 0

3 cos^{2} (x) - 5 + 7 sin (x) = 0 : x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

3 cos^{2} (x) - 5 - 7 sin (x) = 0 : x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

Combine all the solutions

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn, x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

- 10 sin (2 x) = 0

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

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

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

- sin (θ) = \frac{1}{2}

What is the general solution for 3cos(x)cot(x)+7=5csc(x) ?
The general solution for 3cos(x)cot(x)+7=5csc(x) is x=0.33983…+2pin,x=pi-0.33983…+2pin