What is the general solution for 3sin(a)+cos(a)=1 ?

The general solution for 3sin(a)+cos(a)=1 is a=2.49809…+2pin,a=2pin

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Popular Trigonometry >

3sin(a)+cos(a)=1

Solution

3 sin (a) + cos (a) = 1

Solution

a = 2.49809 \dots + 2 πn, a = 2 πn

Degrees

a = 143.13010 \dots^{\circ} + 36 0^{\circ} n, a = 0^{\circ} + 36 0^{\circ} n

Solution steps

3 sin (a) + cos (a) = 1

Subtract

cos (a)

from both sides

3 sin (a) = 1 - cos (a)

Square both sides

(3 sin (a))^{2} = (1 - cos (a))^{2}

Subtract

(1 - cos (a))^{2}

from both sides

9 sin^{2} (a) - 1 + 2 cos (a) - cos^{2} (a) = 0

Rewrite using trig identities

- 1 - cos^{2} (a) + 2 cos (a) + 9 sin^{2} (a)

Use the Pythagorean identity:

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

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

= - 1 - cos^{2} (a) + 2 cos (a) + 9 (1 - cos^{2} (a))

Simplify

- 1 - cos^{2} (a) + 2 cos (a) + 9 (1 - cos^{2} (a)) : 2 cos (a) - 10 cos^{2} (a) + 8

- 1 - cos^{2} (a) + 2 cos (a) + 9 (1 - cos^{2} (a))

Expand

9 (1 - cos^{2} (a)) : 9 - 9 cos^{2} (a)

9 (1 - cos^{2} (a))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = cos^{2} (a)

= 9 \cdot 1 - 9 cos^{2} (a)

Multiply the numbers:

9 \cdot 1 = 9

= 9 - 9 cos^{2} (a)

= - 1 - cos^{2} (a) + 2 cos (a) + 9 - 9 cos^{2} (a)

Simplify

- 1 - cos^{2} (a) + 2 cos (a) + 9 - 9 cos^{2} (a) : 2 cos (a) - 10 cos^{2} (a) + 8

- 1 - cos^{2} (a) + 2 cos (a) + 9 - 9 cos^{2} (a)

Group like terms

= - cos^{2} (a) + 2 cos (a) - 9 cos^{2} (a) - 1 + 9

Add similar elements:

- cos^{2} (a) - 9 cos^{2} (a) = - 10 cos^{2} (a)

= - 10 cos^{2} (a) + 2 cos (a) - 1 + 9

Add/Subtract the numbers:

- 1 + 9 = 8

= 2 cos (a) - 10 cos^{2} (a) + 8

= 2 cos (a) - 10 cos^{2} (a) + 8

= 2 cos (a) - 10 cos^{2} (a) + 8

8 - 10 cos^{2} (a) + 2 cos (a) = 0

Solve by substitution

8 - 10 cos^{2} (a) + 2 cos (a) = 0

Let:

cos (a) = u

8 - 10 u^{2} + 2 u = 0

8 - 10 u^{2} + 2 u = 0 : u = - \frac{4}{5}, u = 1

8 - 10 u^{2} + 2 u = 0

Write in the standard form

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

- 10 u^{2} + 2 u + 8 = 0

Solve with the quadratic formula

- 10 u^{2} + 2 u + 8 = 0

Quadratic Equation Formula:

For

a = - 10, b = 2, c = 8

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 10 ) \cdot 8}{2 ( - 10 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 10 ) \cdot 8}{2 ( - 10 )}

2^{2} - 4 (- 10) \cdot 8 = 18

2^{2} - 4 (- 10) \cdot 8

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 10 \cdot 8

Multiply the numbers:

4 \cdot 10 \cdot 8 = 320

= 2^{2} + 320

2^{2} = 4

= 4 + 320

Add the numbers:

4 + 320 = 324

= 324

Factor the number:

324 = 1 8^{2}

= 1 8^{2}

Apply radical rule:

n a^{n} = a

1 8^{2} = 18

= 18

u_{1, 2} = \frac{- 2 \pm 18}{2 ( - 10 )}

Separate the solutions

u_{1} = \frac{- 2 + 18}{2 ( - 10 )}, u_{2} = \frac{- 2 - 18}{2 ( - 10 )}

u = \frac{- 2 + 18}{2 ( - 10 )} : - \frac{4}{5}

\frac{- 2 + 18}{2 ( - 10 )}

Remove parentheses:

(- a) = - a

= \frac{- 2 + 18}{- 2 \cdot 10}

Add/Subtract the numbers:

- 2 + 18 = 16

= \frac{16}{- 2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{16}{- 20}

Apply the fraction rule:

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

= - \frac{16}{20}

Cancel the common factor:

4

= - \frac{4}{5}

u = \frac{- 2 - 18}{2 ( - 10 )} : 1

\frac{- 2 - 18}{2 ( - 10 )}

Remove parentheses:

(- a) = - a

= \frac{- 2 - 18}{- 2 \cdot 10}

Subtract the numbers:

- 2 - 18 = - 20

= \frac{- 20}{- 2 \cdot 10}

Multiply the numbers:

2 \cdot 10 = 20

= \frac{- 20}{- 20}

Apply the fraction rule:

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

= \frac{20}{20}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - \frac{4}{5}, u = 1

Substitute back

u = cos (a)

cos (a) = - \frac{4}{5}, cos (a) = 1

cos (a) = - \frac{4}{5}, cos (a) = 1

cos (a) = - \frac{4}{5} : a = arccos (- \frac{4}{5}) + 2 πn, a = - arccos (- \frac{4}{5}) + 2 πn

cos (a) = - \frac{4}{5}

Apply trig inverse properties

cos (a) = - \frac{4}{5}

General solutions for

cos (a) = - \frac{4}{5}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

a = arccos (- \frac{4}{5}) + 2 πn, a = - arccos (- \frac{4}{5}) + 2 πn

a = arccos (- \frac{4}{5}) + 2 πn, a = - arccos (- \frac{4}{5}) + 2 πn

cos (a) = 1 : a = 2 πn

cos (a) = 1

General solutions for

cos (a) = 1

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}

a = 0 + 2 πn

a = 0 + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

Combine all the solutions

a = arccos (- \frac{4}{5}) + 2 πn, a = - arccos (- \frac{4}{5}) + 2 πn, a = 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

3 sin (a) + cos (a) = 1

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

Check the solution

arccos (- \frac{4}{5}) + 2 πn :

True

arccos (- \frac{4}{5}) + 2 πn

Plug in

n = 1

arccos (- \frac{4}{5}) + 2 π 1

For

3 sin (a) + cos (a) = 1

plug in

a = arccos (- \frac{4}{5}) + 2 π 1

3 sin (arccos (- \frac{4}{5}) + 2 π 1) + cos (arccos (- \frac{4}{5}) + 2 π 1) = 1

Refine

1 = 1

\Rightarrow True

Check the solution

- arccos (- \frac{4}{5}) + 2 πn :

False

- arccos (- \frac{4}{5}) + 2 πn

Plug in

n = 1

- arccos (- \frac{4}{5}) + 2 π 1

For

3 sin (a) + cos (a) = 1

plug in

a = - arccos (- \frac{4}{5}) + 2 π 1

3 sin (- arccos (- \frac{4}{5}) + 2 π 1) + cos (- arccos (- \frac{4}{5}) + 2 π 1) = 1

Refine

- 2.6 = 1

\Rightarrow False

Check the solution

2 πn :

True

2 πn

Plug in

n = 1

2 π 1

For

3 sin (a) + cos (a) = 1

plug in

a = 2 π 1

3 sin (2 π 1) + cos (2 π 1) = 1

Refine

1 = 1

\Rightarrow True

a = arccos (- \frac{4}{5}) + 2 πn, a = 2 πn

Show solutions in decimal form

a = 2.49809 \dots + 2 πn, a = 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(a)+cos(a)=1 ?
The general solution for 3sin(a)+cos(a)=1 is a=2.49809…+2pin,a=2pin