What is the general solution for 2sin^2(45-a)=1-sin^2(a) ?

The general solution for 2sin^2(45-a)=1-sin^2(a) is a=360n,a=180+360n,a=1.10714…+180n

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2sin^2(45-a)=1-sin^2(a)

Solution

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

Solution

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

Radians

a = 0 + 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

Solution steps

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

Rewrite using trig identities

2 sin^{2} (4 5^{\circ} - a) = 1 - sin^{2} (a)

Rewrite using trig identities

sin (4 5^{\circ} - a)

Use the Angle Difference identity:

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

= sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

Simplify

sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a) : \frac{2 cos ( a ) - 2 sin ( a )}{2}

sin (4 5^{\circ}) cos (a) - cos (4 5^{\circ}) sin (a)

sin (4 5^{\circ}) cos (a) = \frac{2 cos ( a )}{2}

sin (4 5^{\circ}) cos (a)

Simplify

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} cos (a)

Multiply fractions:

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

= \frac{2 cos ( a )}{2}

cos (4 5^{\circ}) sin (a) = \frac{2 sin ( a )}{2}

cos (4 5^{\circ}) sin (a)

Simplify

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} sin (a)

Multiply fractions:

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

= \frac{2 sin ( a )}{2}

= \frac{2 cos ( a )}{2} - \frac{2 sin ( a )}{2}

Apply rule

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

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

= \frac{2 cos ( a ) - 2 sin ( a )}{2}

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

Simplify

2 (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2} : (cos (a) - sin (a))^{2}

2 (\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

(\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2} = \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

(\frac{2 cos ( a ) - 2 sin ( a )}{2})^{2}

\frac{2 cos ( a ) - 2 sin ( a )}{2} = \frac{cos ( a ) - sin ( a )}{2}

\frac{2 cos ( a ) - 2 sin ( a )}{2}

Factor out common term

2

= \frac{2 ( cos ( a ) - sin ( a ) )}{2}

Cancel

\frac{2 ( cos ( a ) - sin ( a ) )}{2} : \frac{cos ( a ) - sin ( a )}{2}

\frac{2 ( cos ( a ) - sin ( a ) )}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

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

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

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

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{cos ( a ) - sin ( a )}{2}

= \frac{cos ( a ) - sin ( a )}{2}

= (\frac{cos ( a ) - sin ( a )}{2})^{2}

Apply exponent rule:

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

= \frac{( cos ( a ) - sin ( a ) ) ^{2}}{( 2 ) ^{2}}

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

= 2 \cdot \frac{( cos ( a ) - sin ( a ) ) ^{2}}{2}

Multiply fractions:

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

= \frac{( cos ( a ) - sin ( a ) ) ^{2} \cdot 2}{2}

Cancel the common factor:

2

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

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

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

Subtract

1 - sin^{2} (a)

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Factor

sin^{2} (a) - 2 cos (a) sin (a) : sin (a) (sin (a) - 2 cos (a))

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

Apply exponent rule:

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

sin^{2} (a) = sin (a) sin (a)

= sin (a) sin (a) - 2 sin (a) cos (a)

Factor out common term

sin (a)

= sin (a) (sin (a) - 2 cos (a))

sin (a) (sin (a) - 2 cos (a)) = 0

Solving each part separately

sin (a) = 0 or sin (a) - 2 cos (a) = 0

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

sin (a) = 0

General solutions for

sin (a) = 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}

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

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

Solve

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

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

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

a = 36 0^{\circ} n

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

sin (a) - 2 cos (a) = 0 : a = arctan (2) + 18 0^{\circ} n

sin (a) - 2 cos (a) = 0

Rewrite using trig identities

sin (a) - 2 cos (a) = 0

Divide both sides by

cos (a), cos (a) \neq = 0

\frac{sin ( a ) - 2 cos ( a )}{cos ( a )} = \frac{0}{cos ( a )}

Simplify

\frac{sin ( a )}{cos ( a )} - 2 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (a) - 2 = 0

tan (a) - 2 = 0

Move

2

to the right side

tan (a) - 2 = 0

Add

2

to both sides

tan (a) - 2 + 2 = 0 + 2

Simplify

tan (a) = 2

tan (a) = 2

Apply trig inverse properties

tan (a) = 2

General solutions for

tan (a) = 2

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

a = arctan (2) + 18 0^{\circ} n

a = arctan (2) + 18 0^{\circ} n

Combine all the solutions

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = arctan (2) + 18 0^{\circ} n

Show solutions in decimal form

a = 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 1.10714 \dots + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(45-a)=1-sin^2(a) ?
The general solution for 2sin^2(45-a)=1-sin^2(a) is a=360n,a=180+360n,a=1.10714…+180n