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Popular >

(sin^2(x)-2sin(x)+1)/(sin(x)-1)=sin(-1)

Solution

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

x = 0.15920 \dots + 2 πn, x = π - 0.15920 \dots + 2 πn

Degrees

x = 9.12152 \dots^{\circ} + 36 0^{\circ} n, x = 170.87847 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

sin (- 1) = - sin (1)

sin (- 1)

Use the following property:

sin (- x) = - sin (x)

sin (- 1) = - sin (1)

= - sin (1)

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

Simplify

\frac{sin ^{2} ( x ) - 2 sin ( x ) + 1}{sin ( x ) - 1} : sin (x) - 1

\frac{sin ^{2} ( x ) - 2 sin ( x ) + 1}{sin ( x ) - 1}

Factor

sin^{2} (x) - 2 sin (x) + 1 : (sin (x) - 1)^{2}

sin^{2} (x) - 2 sin (x) + 1

Rewrite

sin^{2} (x) - 2 sin (x) + 1

sin^{2} (x) - 2 sin (x) \cdot 1 + 1^{2}

sin^{2} (x) - 2 sin (x) + 1

Rewrite

1

1^{2}

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

Rewrite

2 sin (x)

2 sin (x) \cdot 1

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

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

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = sin (x), b = 1

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

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

Cancel the common factor:

sin (x) - 1

= sin (x) - 1

sin (x) - 1 = - sin (1)

Move

1

to the right side

sin (x) - 1 = - sin (1)

Add

1

to both sides

sin (x) - 1 + 1 = - sin (1) + 1

Simplify

sin (x) = - sin (1) + 1

sin (x) = - sin (1) + 1

Apply trig inverse properties

sin (x) = - sin (1) + 1

General solutions for

sin (x) = - sin (1) + 1

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

x = arcsin (- sin (1) + 1) + 2 πn, x = π - arcsin (- sin (1) + 1) + 2 πn

x = arcsin (- sin (1) + 1) + 2 πn, x = π - arcsin (- sin (1) + 1) + 2 πn

Show solutions in decimal form

x = 0.15920 \dots + 2 πn, x = π - 0.15920 \dots + 2 πn

tan^{2} (x) = 3 - 2 tan (x), 0 \leq x \leq 2 π

4 sin (θ) = 4 cos (θ)

sin (θ - 4 0^{\circ}) = 1

2 cos (x) + 2 sin (x) = 2

sin (x) = sin (- x) + 1