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0=sqrt(((10cos(x))-8)^2((10sin(x))-6)^2)

Solution

0 = ((10 cos (x)) - 8)^{2} ((10 sin (x)) - 6)^{2}

Solution

x = 2 π - 0.64350 \dots + 2 πn, x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn

Degrees

x = 323.13010 \dots^{\circ} + 36 0^{\circ} n, x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 143.13010 \dots^{\circ} + 36 0^{\circ} n

Solution steps

0 = ((10 cos (x)) - 8)^{2} ((10 sin (x)) - 6)^{2}

Switch sides

(10 cos (x) - 8)^{2} (10 sin (x) - 6)^{2} = 0

Factor

(10 cos (x) - 8)^{2} (10 sin (x) - 6)^{2} : 4 (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

(10 cos (x) - 8)^{2} (10 sin (x) - 6)^{2}

(10 cos (x) - 8)^{2} = (2 (5 cos (x) - 4))^{2}

(10 cos (x) - 8)^{2}

Factor

10 cos (x) - 8 : 2 (5 cos (x) - 4)

10 cos (x) - 8

Factor out common term

2 : 2 (5 cos (x) - 4)

10 cos (x) - 8

Rewrite

8

2 \cdot 4

Rewrite

10

2 \cdot 5

= 2 \cdot 5 cos (x) - 2 \cdot 4

Factor out common term

2

= 2 (5 cos (x) - 4)

= 2 (5 cos (x) - 4)

= (2 (5 cos (x) - 4))^{2}

= (2 (5 cos (x) - 4))^{2} (10 sin (x) - 6)^{2}

(10 sin (x) - 6)^{2} = (2 (5 sin (x) - 3))^{2}

(10 sin (x) - 6)^{2}

Factor

10 sin (x) - 6 : 2 (5 sin (x) - 3)

10 sin (x) - 6

Factor out common term

2 : 2 (5 sin (x) - 3)

10 sin (x) - 6

Rewrite

6

2 \cdot 3

Rewrite

10

2 \cdot 5

= 2 \cdot 5 sin (x) - 2 \cdot 3

Factor out common term

2

= 2 (5 sin (x) - 3)

= 2 (5 sin (x) - 3)

= (2 (5 sin (x) - 3))^{2}

= (2 (5 cos (x) - 4))^{2} (2 (5 sin (x) - 3))^{2}

(2 (5 cos (x) - 4))^{2} = 2^{2} (5 cos (x) - 4)^{2}

(2 (5 cos (x) - 4))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (5 cos (x) - 4)^{2}

= 2^{2} (5 cos (x) - 4)^{2} (2 (5 sin (x) - 3))^{2}

(2 (5 sin (x) - 3))^{2} = 2^{2} (5 sin (x) - 3)^{2}

(2 (5 sin (x) - 3))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (5 sin (x) - 3)^{2}

= 2^{2} \cdot 2^{2} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

Simplify

2^{2} (5 cos (x) - 4)^{2} \cdot 2^{2} (5 sin (x) - 3)^{2} : 2^{4} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

2^{2} (5 cos (x) - 4)^{2} \cdot 2^{2} (5 sin (x) - 3)^{2}

Apply exponent rule:

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

2^{2} \cdot 2^{2} = 2^{2 + 2}

= (5 cos (x) - 4)^{2} \cdot 2^{2 + 2} (5 sin (x) - 3)^{2}

Add the numbers:

2 + 2 = 4

= (5 cos (x) - 4)^{2} \cdot 2^{4} (5 sin (x) - 3)^{2}

= 2^{4} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 2^{4} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

Apply exponent rule:

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

2^{4} = 2^{2 \cdot 2} = (2^{2})^{2}

= (5 cos (x) - 4)^{2} (2^{2})^{2} (5 sin (x) - 3)^{2}

Apply radical rule:

assuming

a \geq 0

(2^{2})^{2} = 2^{2}

= 2^{2} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

2^{2} = 4

= 4 (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

4 (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2} = 0

Solving each part separately

(5 cos (x) - 4)^{2} = 0 or (5 sin (x) - 3)^{2} = 0

(5 cos (x) - 4)^{2} = 0 : x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

(5 cos (x) - 4)^{2} = 0

Square both sides:

(5 cos (x) - 4)^{2} = 0

(5 cos (x) - 4)^{2} = 0

((5 cos (x) - 4)^{2})^{2} = 0^{2}

Expand

((5 cos (x) - 4)^{2})^{2} : (5 cos (x) - 4)^{2}

((5 cos (x) - 4)^{2})^{2}

Apply radical rule:

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

= (((5 cos (x) - 4)^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= ((5 cos (x) - 4)^{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

= (5 cos (x) - 4)^{2}

Expand

0^{2} : 0

0^{2}

Apply rule

0^{a} = 0

= 0

(5 cos (x) - 4)^{2} = 0

(5 cos (x) - 4)^{2} = 0

Solve

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

(5 cos (x) - 4)^{2} = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

Solve

5 cos (x) - 4 = 0 : cos (x) = \frac{4}{5}

5 cos (x) - 4 = 0

Move

4

to the right side

5 cos (x) - 4 = 0

Add

4

to both sides

5 cos (x) - 4 + 4 = 0 + 4

Simplify

5 cos (x) = 4

5 cos (x) = 4

Divide both sides by

5

5 cos (x) = 4

Divide both sides by

5

\frac{5 cos ( x )}{5} = \frac{4}{5}

Simplify

cos (x) = \frac{4}{5}

cos (x) = \frac{4}{5}

The solution to the quadratic equation is:

cos (x) = \frac{4}{5}

cos (x) = \frac{4}{5}

Verify Solutions:

cos (x) = \frac{4}{5}

True

Check the solutions by plugging them into

(5 cos (x) - 4)^{2} = 0

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

Plug in

cos (x) = \frac{4}{5} :

True

(5 (\frac{4}{5}) - 4)^{2} = 0

(5 (\frac{4}{5}) - 4)^{2} = 0

(5 (\frac{4}{5}) - 4)^{2}

Remove parentheses:

(a) = a

= (5 \cdot \frac{4}{5} - 4)^{2}

(5 \cdot \frac{4}{5} - 4)^{2} = 0

(5 \cdot \frac{4}{5} - 4)^{2}

5 \cdot \frac{4}{5} = 4

5 \cdot \frac{4}{5}

Multiply fractions:

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

= \frac{4 \cdot 5}{5}

Cancel the common factor:

5

= 4

= (4 - 4)^{2}

Subtract the numbers:

4 - 4 = 0

= 0^{2}

Apply rule

0^{a} = 0

= 0

= 0

Apply rule

0 = 0

= 0

0 = 0

True

The solution is

cos (x) = \frac{4}{5}

Apply trig inverse properties

cos (x) = \frac{4}{5}

General solutions for

cos (x) = \frac{4}{5}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

(5 sin (x) - 3)^{2} = 0 : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

(5 sin (x) - 3)^{2} = 0

Square both sides:

(5 sin (x) - 3)^{2} = 0

(5 sin (x) - 3)^{2} = 0

((5 sin (x) - 3)^{2})^{2} = 0^{2}

Expand

((5 sin (x) - 3)^{2})^{2} : (5 sin (x) - 3)^{2}

((5 sin (x) - 3)^{2})^{2}

Apply radical rule:

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

= (((5 sin (x) - 3)^{2})^{\frac{1}{2}})^{2}

Apply exponent rule:

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

= ((5 sin (x) - 3)^{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

= (5 sin (x) - 3)^{2}

Expand

0^{2} : 0

0^{2}

Apply rule

0^{a} = 0

= 0

(5 sin (x) - 3)^{2} = 0

(5 sin (x) - 3)^{2} = 0

Solve

(5 sin (x) - 3)^{2} = 0 : sin (x) = \frac{3}{5}

(5 sin (x) - 3)^{2} = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

Solve

5 sin (x) - 3 = 0 : sin (x) = \frac{3}{5}

5 sin (x) - 3 = 0

Move

3

to the right side

5 sin (x) - 3 = 0

Add

3

to both sides

5 sin (x) - 3 + 3 = 0 + 3

Simplify

5 sin (x) = 3

5 sin (x) = 3

Divide both sides by

5

5 sin (x) = 3

Divide both sides by

5

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

Simplify

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

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

The solution to the quadratic equation is:

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

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

Verify Solutions:

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

True

Check the solutions by plugging them into

(5 sin (x) - 3)^{2} = 0

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

Plug in

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

True

(5 (\frac{3}{5}) - 3)^{2} = 0

(5 (\frac{3}{5}) - 3)^{2} = 0

(5 (\frac{3}{5}) - 3)^{2}

Remove parentheses:

(a) = a

= (5 \cdot \frac{3}{5} - 3)^{2}

(5 \cdot \frac{3}{5} - 3)^{2} = 0

(5 \cdot \frac{3}{5} - 3)^{2}

5 \cdot \frac{3}{5} = 3

5 \cdot \frac{3}{5}

Multiply fractions:

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

= \frac{3 \cdot 5}{5}

Cancel the common factor:

5

= 3

= (3 - 3)^{2}

Subtract the numbers:

3 - 3 = 0

= 0^{2}

Apply rule

0^{a} = 0

= 0

= 0

Apply rule

0 = 0

= 0

0 = 0

True

The solution is

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Combine all the solutions

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn, x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Merge Overlapping Intervals

x = 2 π - arccos (\frac{4}{5}) + 2 πn, x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Show solutions in decimal form

x = 2 π - 0.64350 \dots + 2 πn, x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn

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