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Popular >

d^2+13d+36=(sin^2(x))/2

Solution

d^{2} + 13 d + 36 = \frac{sin ^{2} ( x )}{2}

x = arcsin (2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (2 d^{2} + 26 d + 72) + 2 πn

Solution steps

d^{2} + 13 d + 36 = \frac{sin ^{2} ( x )}{2}

Switch sides

\frac{sin ^{2} ( x )}{2} = d^{2} + 13 d + 36

Solve by substitution

\frac{sin ^{2} ( x )}{2} = d^{2} + 13 d + 36

Let:

sin (x) = u

\frac{u ^{2}}{2} = d^{2} + 13 d + 36

\frac{u ^{2}}{2} = d^{2} + 13 d + 36 : u = 2 d^{2} + 26 d + 72, u = - 2 d^{2} + 26 d + 72

\frac{u ^{2}}{2} = d^{2} + 13 d + 36

Multiply both sides by

2

\frac{u ^{2}}{2} = d^{2} + 13 d + 36

Multiply both sides by

2

\frac{2 u ^{2}}{2} = 2 d^{2} + 2 \cdot 13 d + 2 \cdot 36

Simplify

u^{2} = 2 d^{2} + 26 d + 72

u^{2} = 2 d^{2} + 26 d + 72

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 2 d^{2} + 26 d + 72, u = - 2 d^{2} + 26 d + 72

Substitute back

u = sin (x)

sin (x) = 2 d^{2} + 26 d + 72, sin (x) = - 2 d^{2} + 26 d + 72

sin (x) = 2 d^{2} + 26 d + 72, sin (x) = - 2 d^{2} + 26 d + 72

sin (x) = 2 d^{2} + 26 d + 72 : x = arcsin (2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (- 2 d^{2} + 26 d + 72) + 2 πn

sin (x) = 2 d^{2} + 26 d + 72

Apply trig inverse properties

sin (x) = 2 d^{2} + 26 d + 72

General solutions for

sin (x) = 2 d^{2} + 26 d + 72

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

x = arcsin (2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (- 2 d^{2} + 26 d + 72) + 2 πn

x = arcsin (2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (- 2 d^{2} + 26 d + 72) + 2 πn

sin (x) = - 2 d^{2} + 26 d + 72 : x = arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (2 d^{2} + 26 d + 72) + 2 πn

sin (x) = - 2 d^{2} + 26 d + 72

Apply trig inverse properties

sin (x) = - 2 d^{2} + 26 d + 72

General solutions for

sin (x) = - 2 d^{2} + 26 d + 72

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

x = arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (2 d^{2} + 26 d + 72) + 2 πn

x = arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (2 d^{2} + 26 d + 72) + 2 πn

Combine all the solutions

x = arcsin (2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = arcsin (- 2 d^{2} + 26 d + 72) + 2 πn, x = π + arcsin (2 d^{2} + 26 d + 72) + 2 πn

\frac{tan ^{2} ( x ) - 4}{cos ( x ) + 5} = 0

cot^{2} (x) = sec^{2} (x) - 1

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

7 sin^{2} (x) + 2 sin^{2} (x) - 3 cos^{2} (x) = 0

sin^{2} (x) - cos^{2} (x) = cos^{4} (x)