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Popular >

sin^2(x)+1= 7/2 cos^2(x)

Solution

sin^{2} (x) + 1 = \frac{7}{2} cos^{2} (x)

Solution

x = 0.84106 \dots + 2 πn, x = π - 0.84106 \dots + 2 πn, x = - 0.84106 \dots + 2 πn, x = π + 0.84106 \dots + 2 πn

Degrees

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 228.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) + 1 = \frac{7}{2} cos^{2} (x)

Subtract

\frac{7}{2} cos^{2} (x)

from both sides

sin^{2} (x) + 1 - \frac{7}{2} cos^{2} (x) = 0

Simplify

sin^{2} (x) + 1 - \frac{7}{2} cos^{2} (x) : \frac{2 sin ^{2} ( x ) + 2 - 7 cos ^{2} ( x )}{2}

sin^{2} (x) + 1 - \frac{7}{2} cos^{2} (x)

Multiply

\frac{7}{2} cos^{2} (x) : \frac{7 cos ^{2} ( x )}{2}

\frac{7}{2} cos^{2} (x)

Multiply fractions:

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

= \frac{7 cos ^{2} ( x )}{2}

= sin^{2} (x) + 1 - \frac{7 cos ^{2} ( x )}{2}

Convert element to fraction:

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

= \frac{sin ^{2} ( x ) \cdot 2}{2} + \frac{1 \cdot 2}{2} - \frac{7 cos ^{2} ( x )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( x ) \cdot 2 + 1 \cdot 2 - 7 cos ^{2} ( x )}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 sin ^{2} ( x ) + 2 - 7 cos ^{2} ( x )}{2}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= 2 + 2 sin^{2} (x) - 7 (1 - sin^{2} (x))

Simplify

2 + 2 sin^{2} (x) - 7 (1 - sin^{2} (x)) : 9 sin^{2} (x) - 5

2 + 2 sin^{2} (x) - 7 (1 - sin^{2} (x))

Expand

- 7 (1 - sin^{2} (x)) : - 7 + 7 sin^{2} (x)

- 7 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 7, b = 1, c = sin^{2} (x)

= - 7 \cdot 1 - (- 7) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 7 \cdot 1 + 7 sin^{2} (x)

Multiply the numbers:

7 \cdot 1 = 7

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

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

Simplify

2 + 2 sin^{2} (x) - 7 + 7 sin^{2} (x) : 9 sin^{2} (x) - 5

2 + 2 sin^{2} (x) - 7 + 7 sin^{2} (x)

Group like terms

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

Add similar elements:

2 sin^{2} (x) + 7 sin^{2} (x) = 9 sin^{2} (x)

= 9 sin^{2} (x) + 2 - 7

Add/Subtract the numbers:

2 - 7 = - 5

= 9 sin^{2} (x) - 5

= 9 sin^{2} (x) - 5

= 9 sin^{2} (x) - 5

- 5 + 9 sin^{2} (x) = 0

Solve by substitution

- 5 + 9 sin^{2} (x) = 0

Let:

sin (x) = u

- 5 + 9 u^{2} = 0

- 5 + 9 u^{2} = 0 : u = \frac{5}{3}, u = - \frac{5}{3}

- 5 + 9 u^{2} = 0

Move

5

to the right side

- 5 + 9 u^{2} = 0

Add

5

to both sides

- 5 + 9 u^{2} + 5 = 0 + 5

Simplify

9 u^{2} = 5

9 u^{2} = 5

Divide both sides by

9

9 u^{2} = 5

Divide both sides by

9

\frac{9 u ^{2}}{9} = \frac{5}{9}

Simplify

u^{2} = \frac{5}{9}

u^{2} = \frac{5}{9}

For

x^{2} = f (a)

the solutions are

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

u = \frac{5}{9}, u = - \frac{5}{9}

\frac{5}{9} = \frac{5}{3}

\frac{5}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{5}{3}

- \frac{5}{9} = - \frac{5}{3}

- \frac{5}{9}

Simplify

\frac{5}{9} : \frac{5}{3}

\frac{5}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= \frac{5}{3}

= - \frac{5}{3}

u = \frac{5}{3}, u = - \frac{5}{3}

Substitute back

u = sin (x)

sin (x) = \frac{5}{3}, sin (x) = - \frac{5}{3}

sin (x) = \frac{5}{3}, sin (x) = - \frac{5}{3}

sin (x) = \frac{5}{3} : x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn

sin (x) = \frac{5}{3}

Apply trig inverse properties

sin (x) = \frac{5}{3}

General solutions for

sin (x) = \frac{5}{3}

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

x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn

x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn

sin (x) = - \frac{5}{3} : x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

sin (x) = - \frac{5}{3}

Apply trig inverse properties

sin (x) = - \frac{5}{3}

General solutions for

sin (x) = - \frac{5}{3}

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

x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

Combine all the solutions

x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn, x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

Show solutions in decimal form

x = 0.84106 \dots + 2 πn, x = π - 0.84106 \dots + 2 πn, x = - 0.84106 \dots + 2 πn, x = π + 0.84106 \dots + 2 πn

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