What is the general solution for 1=sin(90-θ)-0.075cos(90-θ) ?

The general solution for 1=sin(90-θ)-0.075cos(90-θ) is θ=-0.14971…+360n,θ=360n

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1=sin(90-θ)-0.075cos(90-θ)

Solution

1 = sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ)

Solution

θ = - 0.14971 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} n

Radians

θ = - 0.14971 \dots + 2 πn, θ = 0 + 2 πn

Solution steps

1 = sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ)

Add

0.075 cos (9 0^{\circ} - θ)

to both sides

sin (9 0^{\circ} - θ) = 1 + 0.075 cos (9 0^{\circ} - θ)

Square both sides

sin^{2} (9 0^{\circ} - θ) = (1 + 0.075 cos (9 0^{\circ} - θ))^{2}

Rewrite using trig identities

sin^{2} (9 0^{\circ} - θ) = (1 + 0.075 cos (9 0^{\circ} - θ))^{2}

Rewrite using trig identities

sin (9 0^{\circ} - θ)

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (9 0^{\circ}) cos (θ) - cos (9 0^{\circ}) sin (θ)

Simplify

sin (9 0^{\circ}) cos (θ) - cos (9 0^{\circ}) sin (θ) : cos (θ)

sin (9 0^{\circ}) cos (θ) - cos (9 0^{\circ}) sin (θ)

sin (9 0^{\circ}) cos (θ) = cos (θ)

sin (9 0^{\circ}) cos (θ)

Simplify

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot cos (θ)

Multiply:

1 \cdot cos (θ) = cos (θ)

= cos (θ)

cos (9 0^{\circ}) sin (θ) = 0

cos (9 0^{\circ}) sin (θ)

Simplify

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot sin (θ)

Apply rule

0 \cdot a = 0

= 0

= cos (θ) - 0

cos (θ) - 0 = cos (θ)

= cos (θ)

= cos (θ)

Use the Angle Difference identity:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= cos (9 0^{\circ}) cos (θ) + sin (9 0^{\circ}) sin (θ)

Simplify

cos (9 0^{\circ}) cos (θ) + sin (9 0^{\circ}) sin (θ) : sin (θ)

cos (9 0^{\circ}) cos (θ) + sin (9 0^{\circ}) sin (θ)

cos (9 0^{\circ}) cos (θ) = 0

cos (9 0^{\circ}) cos (θ)

Simplify

cos (9 0^{\circ}) : 0

cos (9 0^{\circ})

Use the following trivial identity:

cos (9 0^{\circ}) = 0

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= 0

= 0 \cdot cos (θ)

Apply rule

0 \cdot a = 0

= 0

sin (9 0^{\circ}) sin (θ) = sin (θ)

sin (9 0^{\circ}) sin (θ)

Simplify

sin (9 0^{\circ}) : 1

sin (9 0^{\circ})

Use the following trivial identity:

sin (9 0^{\circ}) = 1

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= 1 \cdot sin (θ)

Multiply:

1 \cdot sin (θ) = sin (θ)

= sin (θ)

= 0 + sin (θ)

0 + sin (θ) = sin (θ)

= sin (θ)

= sin (θ)

(cos (θ))^{2} = (1 + 0.075 sin (θ))^{2}

Simplify

(cos (θ))^{2} : cos^{2} (θ)

(cos (θ))^{2}

Remove parentheses:

(a) = a

= cos^{2} (θ)

cos^{2} (θ) = (1 + 0.075 sin (θ))^{2}

cos^{2} (θ) = (1 + 0.075 sin (θ))^{2}

Subtract

(1 + 0.075 sin (θ))^{2}

from both sides

cos^{2} (θ) - 1 - 0.15 sin (θ) - 0.005625 sin^{2} (θ) = 0

Rewrite using trig identities

- 1 + cos^{2} (θ) - 0.005625 sin^{2} (θ) - 0.15 sin (θ)

Use the Pythagorean identity:

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

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

= - 0.005625 sin^{2} (θ) - 0.15 sin (θ) - sin^{2} (θ)

Simplify

= - 1.005625 sin^{2} (θ) - 0.15 sin (θ)

- 0.15 sin (θ) - 1.005625 sin^{2} (θ) = 0

Solve by substitution

- 0.15 sin (θ) - 1.005625 sin^{2} (θ) = 0

Let:

sin (θ) = u

- 0.15 u - 1.005625 u^{2} = 0

- 0.15 u - 1.005625 u^{2} = 0 : u = - \frac{0.3}{2.01125}, u = 0

- 0.15 u - 1.005625 u^{2} = 0

Write in the standard form

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

- 1.005625 u^{2} - 0.15 u = 0

Solve with the quadratic formula

- 1.005625 u^{2} - 0.15 u = 0

Quadratic Equation Formula:

For

a = - 1.005625, b = - 0.15, c = 0

u_{1, 2} = \frac{- ( - 0.15 ) \pm ( - 0.15 ) ^{2} - 4 ( - 1.005625 ) \cdot 0}{2 ( - 1.005625 )}

u_{1, 2} = \frac{- ( - 0.15 ) \pm ( - 0.15 ) ^{2} - 4 ( - 1.005625 ) \cdot 0}{2 ( - 1.005625 )}

(- 0.15)^{2} - 4 (- 1.005625) \cdot 0 = 0.15

(- 0.15)^{2} - 4 (- 1.005625) \cdot 0

Apply rule

- (- a) = a

= (- 0.15)^{2} + 4 \cdot 1.005625 \cdot 0

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 0.15)^{2} = 0.1 5^{2}

= 0.1 5^{2} + 4 \cdot 0 \cdot 1.005625

Apply rule

0 \cdot a = 0

= 0.1 5^{2} + 0

0.1 5^{2} + 0 = 0.1 5^{2}

= 0.1 5^{2}

Apply radical rule:

assuming

a \geq 0

= 0.15

u_{1, 2} = \frac{- ( - 0.15 ) \pm 0.15}{2 ( - 1.005625 )}

Separate the solutions

u_{1} = \frac{- ( - 0.15 ) + 0.15}{2 ( - 1.005625 )}, u_{2} = \frac{- ( - 0.15 ) - 0.15}{2 ( - 1.005625 )}

u = \frac{- ( - 0.15 ) + 0.15}{2 ( - 1.005625 )} : - \frac{0.3}{2.01125}

\frac{- ( - 0.15 ) + 0.15}{2 ( - 1.005625 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{0.15 + 0.15}{- 2 \cdot 1.005625}

Add the numbers:

0.15 + 0.15 = 0.3

= \frac{0.3}{- 2 \cdot 1.005625}

Multiply the numbers:

2 \cdot 1.005625 = 2.01125

= \frac{0.3}{- 2.01125}

Apply the fraction rule:

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

= - \frac{0.3}{2.01125}

u = \frac{- ( - 0.15 ) - 0.15}{2 ( - 1.005625 )} : 0

\frac{- ( - 0.15 ) - 0.15}{2 ( - 1.005625 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{0.15 - 0.15}{- 2 \cdot 1.005625}

Subtract the numbers:

0.15 - 0.15 = 0

= \frac{0}{- 2 \cdot 1.005625}

Multiply the numbers:

2 \cdot 1.005625 = 2.01125

= \frac{0}{- 2.01125}

Apply the fraction rule:

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

= - \frac{0}{2.01125}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

The solutions to the quadratic equation are:

u = - \frac{0.3}{2.01125}, u = 0

Substitute back

u = sin (θ)

sin (θ) = - \frac{0.3}{2.01125}, sin (θ) = 0

sin (θ) = - \frac{0.3}{2.01125}, sin (θ) = 0

sin (θ) = - \frac{0.3}{2.01125} : θ = arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} n

sin (θ) = - \frac{0.3}{2.01125}

Apply trig inverse properties

sin (θ) = - \frac{0.3}{2.01125}

General solutions for

sin (θ) = - \frac{0.3}{2.01125}

sin (x) = - a \Rightarrow x = arcsin (- a) + 36 0^{\circ} n, x = 18 0^{\circ} + arcsin (a) + 36 0^{\circ} n

θ = arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} n

θ = arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} n

sin (θ) = 0 : θ = 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

sin (θ) = 0

General solutions for

sin (θ) = 0

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = 0 + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Solve

θ = 0 + 36 0^{\circ} n : θ = 36 0^{\circ} n

θ = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

θ = 36 0^{\circ} n

θ = 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Combine all the solutions

θ = arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n, θ = 18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} n, θ = 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ) = 1

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

Check the solution

arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n :

True

arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n

Plug in

n = 1

arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} 1

For

sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ) = 1

plug in

θ = arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} 1

sin (9 0^{\circ} - (arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} 1)) - 0.075 cos (9 0^{\circ} - (arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} 1)) = 1

Refine

1 = 1

\Rightarrow True

Check the solution

18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} n :

False

18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} n

Plug in

n = 1

18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} 1

For

sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ) = 1

plug in

θ = 18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} 1

sin (9 0^{\circ} - (18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} 1)) - 0.075 cos (9 0^{\circ} - (18 0^{\circ} + arcsin (\frac{0.3}{2.01125}) + 36 0^{\circ} 1)) = 1

Refine

- 0.97762 \dots = 1

\Rightarrow False

Check the solution

36 0^{\circ} n :

True

36 0^{\circ} n

Plug in

n = 1

36 0^{\circ} 1

For

sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ) = 1

plug in

θ = 36 0^{\circ} 1

sin (9 0^{\circ} - 36 0^{\circ} 1) - 0.075 cos (9 0^{\circ} - 36 0^{\circ} 1) = 1

Refine

1 = 1

\Rightarrow True

Check the solution

18 0^{\circ} + 36 0^{\circ} n :

False

18 0^{\circ} + 36 0^{\circ} n

Plug in

n = 1

18 0^{\circ} + 36 0^{\circ} 1

For

sin (9 0^{\circ} - θ) - 0.075 cos (9 0^{\circ} - θ) = 1

plug in

θ = 18 0^{\circ} + 36 0^{\circ} 1

sin (9 0^{\circ} - (18 0^{\circ} + 36 0^{\circ} 1)) - 0.075 cos (9 0^{\circ} - (18 0^{\circ} + 36 0^{\circ} 1)) = 1

Refine

- 1 = 1

\Rightarrow False

θ = arcsin (- \frac{0.3}{2.01125}) + 36 0^{\circ} n, θ = 36 0^{\circ} n

Show solutions in decimal form

θ = - 0.14971 \dots + 36 0^{\circ} n, θ = 36 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 1=sin(90-θ)-0.075cos(90-θ) ?
The general solution for 1=sin(90-θ)-0.075cos(90-θ) is θ=-0.14971…+360n,θ=360n