What is the general solution for sec(y)+5tan(y)=3cos(y) ?

The general solution for sec(y)+5tan(y)=3cos(y) is y=0.33983…+2pin,y=pi-0.33983…+2pin

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Popular Trigonometry >

sec(y)+5tan(y)=3cos(y)

Solution

sec (y) + 5 tan (y) = 3 cos (y)

Solution

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

Degrees

y = 19.47122 \dots^{\circ} + 36 0^{\circ} n, y = 160.52877 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sec (y) + 5 tan (y) = 3 cos (y)

Subtract

3 cos (y)

from both sides

sec (y) + 5 tan (y) - 3 cos (y) = 0

Express with sin, cos

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

Simplify

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

\frac{1}{cos ( y )} + 5 \cdot \frac{sin ( y )}{cos ( y )} - 3 cos (y)

Multiply

5 \cdot \frac{sin ( y )}{cos ( y )} : \frac{5 sin ( y )}{cos ( y )}

5 \cdot \frac{sin ( y )}{cos ( y )}

Multiply fractions:

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

= \frac{sin ( y ) \cdot 5}{cos ( y )}

= \frac{1}{cos ( y )} + \frac{5 sin ( y )}{cos ( y )} - 3 cos (y)

Combine the fractions

\frac{1}{cos ( y )} + \frac{5 sin ( y )}{cos ( y )} : \frac{1 + 5 sin ( y )}{cos ( y )}

Apply rule

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

= \frac{1 + 5 sin ( y )}{cos ( y )}

= \frac{5 sin ( y ) + 1}{cos ( y )} - 3 cos (y)

Convert element to fraction:

3 cos (y) = \frac{3 cos ( y ) cos ( y )}{cos ( y )}

= \frac{1 + sin ( y ) \cdot 5}{cos ( y )} - \frac{3 cos ( y ) cos ( y )}{cos ( y )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + sin ( y ) \cdot 5 - 3 cos ( y ) cos ( y )}{cos ( y )}

1 + sin (y) \cdot 5 - 3 cos (y) cos (y) = 1 + 5 sin (y) - 3 cos^{2} (y)

1 + sin (y) \cdot 5 - 3 cos (y) cos (y)

3 cos (y) cos (y) = 3 cos^{2} (y)

3 cos (y) cos (y)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (y) cos (y) = cos^{1 + 1} (y)

= 3 cos^{1 + 1} (y)

Add the numbers:

1 + 1 = 2

= 3 cos^{2} (y)

= 1 + 5 sin (y) - 3 cos^{2} (y)

= \frac{1 + 5 sin ( y ) - 3 cos ^{2} ( y )}{cos ( y )}

\frac{1 + 5 sin ( y ) - 3 cos ^{2} ( y )}{cos ( y )} = 0

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

1 + 5 sin (y) - 3 cos^{2} (y) = 0

Add

3 cos^{2} (y)

to both sides

1 + 5 sin (y) = 3 cos^{2} (y)

Square both sides

(1 + 5 sin (y))^{2} = (3 cos^{2} (y))^{2}

Subtract

(3 cos^{2} (y))^{2}

from both sides

(1 + 5 sin (y))^{2} - 9 cos^{4} (y) = 0

Factor

(1 + 5 sin (y))^{2} - 9 cos^{4} (y) : (1 + 5 sin (y) + 3 cos^{2} (y)) (1 + 5 sin (y) - 3 cos^{2} (y))

(1 + 5 sin (y))^{2} - 9 cos^{4} (y)

Rewrite

(1 + 5 sin (y))^{2} - 9 cos^{4} (y)

(1 + 5 sin (y))^{2} - (3 cos^{2} (y))^{2}

(1 + 5 sin (y))^{2} - 9 cos^{4} (y)

Rewrite

9

3^{2}

= (1 + 5 sin (y))^{2} - 3^{2} cos^{4} (y)

Apply exponent rule:

a^{b c} = (a^{b})^{c}

cos^{4} (y) = (cos^{2} (y))^{2}

= (1 + 5 sin (y))^{2} - 3^{2} (cos^{2} (y))^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

3^{2} (cos^{2} (y))^{2} = (3 cos^{2} (y))^{2}

= (1 + 5 sin (y))^{2} - (3 cos^{2} (y))^{2}

= (1 + 5 sin (y))^{2} - (3 cos^{2} (y))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(1 + 5 sin (y))^{2} - (3 cos^{2} (y))^{2} = ((1 + 5 sin (y)) + 3 cos^{2} (y)) ((1 + 5 sin (y)) - 3 cos^{2} (y))

= ((1 + 5 sin (y)) + 3 cos^{2} (y)) ((1 + 5 sin (y)) - 3 cos^{2} (y))

Refine

= (3 cos^{2} (y) + 5 sin (y) + 1) (5 sin (y) - 3 cos^{2} (y) + 1)

(1 + 5 sin (y) + 3 cos^{2} (y)) (1 + 5 sin (y) - 3 cos^{2} (y)) = 0

Solving each part separately

1 + 5 sin (y) + 3 cos^{2} (y) = 0 or 1 + 5 sin (y) - 3 cos^{2} (y) = 0

1 + 5 sin (y) + 3 cos^{2} (y) = 0 : y = arcsin (- \frac{- 5 + 73}{6}) + 2 πn, y = π + arcsin (\frac{- 5 + 73}{6}) + 2 πn

1 + 5 sin (y) + 3 cos^{2} (y) = 0

Rewrite using trig identities

1 + 3 cos^{2} (y) + 5 sin (y)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 1 + 3 (1 - sin^{2} (y)) + 5 sin (y)

Simplify

1 + 3 (1 - sin^{2} (y)) + 5 sin (y) : 5 sin (y) - 3 sin^{2} (y) + 4

1 + 3 (1 - sin^{2} (y)) + 5 sin (y)

Expand

3 (1 - sin^{2} (y)) : 3 - 3 sin^{2} (y)

3 (1 - sin^{2} (y))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (y)

= 3 \cdot 1 - 3 sin^{2} (y)

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 sin^{2} (y)

= 1 + 3 - 3 sin^{2} (y) + 5 sin (y)

Add the numbers:

1 + 3 = 4

= 5 sin (y) - 3 sin^{2} (y) + 4

= 5 sin (y) - 3 sin^{2} (y) + 4

4 - 3 sin^{2} (y) + 5 sin (y) = 0

Solve by substitution

4 - 3 sin^{2} (y) + 5 sin (y) = 0

Let:

sin (y) = u

4 - 3 u^{2} + 5 u = 0

4 - 3 u^{2} + 5 u = 0 : u = - \frac{- 5 + 73}{6}, u = \frac{5 + 73}{6}

4 - 3 u^{2} + 5 u = 0

Write in the standard form

a x^{2} + b x + c = 0

- 3 u^{2} + 5 u + 4 = 0

Solve with the quadratic formula

- 3 u^{2} + 5 u + 4 = 0

Quadratic Equation Formula:

For

a = - 3, b = 5, c = 4

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

5^{2} - 4 (- 3) \cdot 4 = 73

5^{2} - 4 (- 3) \cdot 4

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 5^{2} + 48

5^{2} = 25

= 25 + 48

Add the numbers:

25 + 48 = 73

= 73

u_{1, 2} = \frac{- 5 \pm 73}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 5 + 73}{2 ( - 3 )}, u_{2} = \frac{- 5 - 73}{2 ( - 3 )}

u = \frac{- 5 + 73}{2 ( - 3 )} : - \frac{- 5 + 73}{6}

\frac{- 5 + 73}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 5 + 73}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 5 + 73}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 5 + 73}{6}

u = \frac{- 5 - 73}{2 ( - 3 )} : \frac{5 + 73}{6}

\frac{- 5 - 73}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 5 - 73}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 5 - 73}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

- 5 - 73 = - (5 + 73)

= \frac{5 + 73}{6}

The solutions to the quadratic equation are:

u = - \frac{- 5 + 73}{6}, u = \frac{5 + 73}{6}

Substitute back

u = sin (y)

sin (y) = - \frac{- 5 + 73}{6}, sin (y) = \frac{5 + 73}{6}

sin (y) = - \frac{- 5 + 73}{6}, sin (y) = \frac{5 + 73}{6}

sin (y) = - \frac{- 5 + 73}{6} : y = arcsin (- \frac{- 5 + 73}{6}) + 2 πn, y = π + arcsin (\frac{- 5 + 73}{6}) + 2 πn

sin (y) = - \frac{- 5 + 73}{6}

Apply trig inverse properties

sin (y) = - \frac{- 5 + 73}{6}

General solutions for

sin (y) = - \frac{- 5 + 73}{6}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

y = arcsin (- \frac{- 5 + 73}{6}) + 2 πn, y = π + arcsin (\frac{- 5 + 73}{6}) + 2 πn

y = arcsin (- \frac{- 5 + 73}{6}) + 2 πn, y = π + arcsin (\frac{- 5 + 73}{6}) + 2 πn

sin (y) = \frac{5 + 73}{6} :

No Solution

sin (y) = \frac{5 + 73}{6}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

y = arcsin (- \frac{- 5 + 73}{6}) + 2 πn, y = π + arcsin (\frac{- 5 + 73}{6}) + 2 πn

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

1 + 5 sin (y) - 3 cos^{2} (y) = 0

Rewrite using trig identities

1 - 3 cos^{2} (y) + 5 sin (y)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 1 - 3 (1 - sin^{2} (y)) + 5 sin (y)

Simplify

1 - 3 (1 - sin^{2} (y)) + 5 sin (y) : 3 sin^{2} (y) + 5 sin (y) - 2

1 - 3 (1 - sin^{2} (y)) + 5 sin (y)

Expand

- 3 (1 - sin^{2} (y)) : - 3 + 3 sin^{2} (y)

- 3 (1 - sin^{2} (y))

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 1, c = sin^{2} (y)

= - 3 \cdot 1 - (- 3) sin^{2} (y)

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 1 + 3 sin^{2} (y)

Multiply the numbers:

3 \cdot 1 = 3

= - 3 + 3 sin^{2} (y)

= 1 - 3 + 3 sin^{2} (y) + 5 sin (y)

Subtract the numbers:

1 - 3 = - 2

= 3 sin^{2} (y) + 5 sin (y) - 2

= 3 sin^{2} (y) + 5 sin (y) - 2

- 2 + 3 sin^{2} (y) + 5 sin (y) = 0

Solve by substitution

- 2 + 3 sin^{2} (y) + 5 sin (y) = 0

Let:

sin (y) = u

- 2 + 3 u^{2} + 5 u = 0

- 2 + 3 u^{2} + 5 u = 0 : u = \frac{1}{3}, u = - 2

- 2 + 3 u^{2} + 5 u = 0

Write in the standard form

a x^{2} + b x + c = 0

3 u^{2} + 5 u - 2 = 0

Solve with the quadratic formula

3 u^{2} + 5 u - 2 = 0

Quadratic Equation Formula:

For

a = 3, b = 5, c = - 2

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

5^{2} - 4 \cdot 3 (- 2) = 7

5^{2} - 4 \cdot 3 (- 2)

Apply rule

- (- a) = a

= 5^{2} + 4 \cdot 3 \cdot 2

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 5 + 7}{2 \cdot 3}, u_{2} = \frac{- 5 - 7}{2 \cdot 3}

u = \frac{- 5 + 7}{2 \cdot 3} : \frac{1}{3}

\frac{- 5 + 7}{2 \cdot 3}

Add/Subtract the numbers:

- 5 + 7 = 2

= \frac{2}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{6}

Cancel the common factor:

2

= \frac{1}{3}

u = \frac{- 5 - 7}{2 \cdot 3} : - 2

\frac{- 5 - 7}{2 \cdot 3}

Subtract the numbers:

- 5 - 7 = - 12

= \frac{- 12}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{6}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = sin (y)

sin (y) = \frac{1}{3}, sin (y) = - 2

sin (y) = \frac{1}{3}, sin (y) = - 2

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

sin (y) = \frac{1}{3}

Apply trig inverse properties

sin (y) = \frac{1}{3}

General solutions for

sin (y) = \frac{1}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (y) = - 2 :

No Solution

sin (y) = - 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

Combine all the solutions

y = arcsin (- \frac{- 5 + 73}{6}) + 2 πn, y = π + arcsin (\frac{- 5 + 73}{6}) + 2 πn, y = arcsin (\frac{1}{3}) + 2 πn, y = π - arcsin (\frac{1}{3}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sec (y) + 5 tan (y) = 3 cos (y)

Remove the ones that don't agree with the equation.

Check the solution

arcsin (- \frac{- 5 + 73}{6}) + 2 πn :

False

arcsin (- \frac{- 5 + 73}{6}) + 2 πn

Plug in

n = 1

arcsin (- \frac{- 5 + 73}{6}) + 2 π 1

For

sec (y) + 5 tan (y) = 3 cos (y)

plug in

y = arcsin (- \frac{- 5 + 73}{6}) + 2 π 1

sec (arcsin (- \frac{- 5 + 73}{6}) + 2 π 1) + 5 tan (arcsin (- \frac{- 5 + 73}{6}) + 2 π 1) = 3 cos (arcsin (- \frac{- 5 + 73}{6}) + 2 π 1)

Refine

- 2.42074 \dots = 2.42074 \dots

\Rightarrow False

Check the solution

π + arcsin (\frac{- 5 + 73}{6}) + 2 πn :

False

π + arcsin (\frac{- 5 + 73}{6}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{- 5 + 73}{6}) + 2 π 1

For

sec (y) + 5 tan (y) = 3 cos (y)

plug in

y = π + arcsin (\frac{- 5 + 73}{6}) + 2 π 1

sec (π + arcsin (\frac{- 5 + 73}{6}) + 2 π 1) + 5 tan (π + arcsin (\frac{- 5 + 73}{6}) + 2 π 1) = 3 cos (π + arcsin (\frac{- 5 + 73}{6}) + 2 π 1)

Refine

2.42074 \dots = - 2.42074 \dots

\Rightarrow False

Check the solution

arcsin (\frac{1}{3}) + 2 πn :

True

arcsin (\frac{1}{3}) + 2 πn

Plug in

n = 1

arcsin (\frac{1}{3}) + 2 π 1

For

sec (y) + 5 tan (y) = 3 cos (y)

plug in

y = arcsin (\frac{1}{3}) + 2 π 1

sec (arcsin (\frac{1}{3}) + 2 π 1) + 5 tan (arcsin (\frac{1}{3}) + 2 π 1) = 3 cos (arcsin (\frac{1}{3}) + 2 π 1)

Refine

2.82842 \dots = 2.82842 \dots

\Rightarrow True

Check the solution

π - arcsin (\frac{1}{3}) + 2 πn :

True

π - arcsin (\frac{1}{3}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{1}{3}) + 2 π 1

For

sec (y) + 5 tan (y) = 3 cos (y)

plug in

y = π - arcsin (\frac{1}{3}) + 2 π 1

sec (π - arcsin (\frac{1}{3}) + 2 π 1) + 5 tan (π - arcsin (\frac{1}{3}) + 2 π 1) = 3 cos (π - arcsin (\frac{1}{3}) + 2 π 1)

Refine

- 2.82842 \dots = - 2.82842 \dots

\Rightarrow True

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

Show solutions in decimal form

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec(y)+5tan(y)=3cos(y) ?
The general solution for sec(y)+5tan(y)=3cos(y) is y=0.33983…+2pin,y=pi-0.33983…+2pin