Trigonometry Examples

$20 \sin^{2} (x) - 21 \sin (x) + 4 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = \sin (x)$ . Substitute $u$ for all occurrences of $\sin (x)$ .

$20 u^{2} - 21 u + 4 = 0$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 20 \cdot 4 = 80$ and whose sum is $b = - 21$ .

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Step 1.2.1.1

Factor $- 21$ out of $- 21 u$ .

$20 u^{2} - 21 u + 4 = 0$

Step 1.2.1.2

Rewrite $- 21$ as $- 5$ plus $- 16$

$20 u^{2} + (- 5 - 16) u + 4 = 0$

Step 1.2.1.3

Apply the distributive property.

$20 u^{2} - 5 u - 16 u + 4 = 0$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$(20 u^{2} - 5 u) - 16 u + 4 = 0$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$5 u (4 u - 1) - 4 (4 u - 1) = 0$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (5 u - 4) = 0$

Step 1.3

Replace all occurrences of $u$ with $\sin (x)$ .

$(4 \sin (x) - 1) (5 \sin (x) - 4) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 \sin (x) - 1 = 0$

$5 \sin (x) - 4 = 0$

Step 3

Set $4 \sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $4 \sin (x) - 1$ equal to $0$ .

$4 \sin (x) - 1 = 0$

Step 3.2

Solve $4 \sin (x) - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$4 \sin (x) = 1$

Step 3.2.2

Divide each term in $4 \sin (x) = 1$ by $4$ and simplify.

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Step 3.2.2.1

Divide each term in $4 \sin (x) = 1$ by $4$ .

$\frac{4 \sin (x)}{4} = \frac{1}{4}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{4 \sin (x)}{4} = \frac{1}{4}$

Step 3.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{1}{4}$

Step 3.2.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{1}{4})$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Evaluate $\arcsin (\frac{1}{4})$ .

$x = 0.25268025$

Step 3.2.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.25268025$

Step 3.2.6

Solve for $x$ .

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Step 3.2.6.1

Remove parentheses.

$x = 3.14159265 - 0.25268025$

Step 3.2.6.2

Remove parentheses.

$x = (3.14159265) - 0.25268025$

Step 3.2.6.3

Subtract $0.25268025$ from $3.14159265$ .

$x = 2.88891239$

Step 3.2.7

Find the period of $\sin (x)$ .

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Step 3.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2.8

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.25268025 + 2 π n, 2.88891239 + 2 π n$ , for any integer $n$

Step 4

Set $5 \sin (x) - 4$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $5 \sin (x) - 4$ equal to $0$ .

$5 \sin (x) - 4 = 0$

Step 4.2

Solve $5 \sin (x) - 4 = 0$ for $x$ .

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Step 4.2.1

Add $4$ to both sides of the equation.

$5 \sin (x) = 4$

Step 4.2.2

Divide each term in $5 \sin (x) = 4$ by $5$ and simplify.

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Step 4.2.2.1

Divide each term in $5 \sin (x) = 4$ by $5$ .

$\frac{5 \sin (x)}{5} = \frac{4}{5}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{5 \sin (x)}{5} = \frac{4}{5}$

Step 4.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{4}{5}$

Step 4.2.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{4}{5})$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Evaluate $\arcsin (\frac{4}{5})$ .

$x = 0.92729521$

Step 4.2.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.92729521$

Step 4.2.6

Solve for $x$ .

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Step 4.2.6.1

Remove parentheses.

$x = 3.14159265 - 0.92729521$

Step 4.2.6.2

Remove parentheses.

$x = (3.14159265) - 0.92729521$

Step 4.2.6.3

Subtract $0.92729521$ from $3.14159265$ .

$x = 2.21429743$

Step 4.2.7

Find the period of $\sin (x)$ .

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Step 4.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.8

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.92729521 + 2 π n, 2.21429743 + 2 π n$ , for any integer $n$

Step 5

The final solution is all the values that make $(4 \sin (x) - 1) (5 \sin (x) - 4) = 0$ true.

$x = 0.25268025 + 2 π n, 2.88891239 + 2 π n, 0.92729521 + 2 π n, 2.21429743 + 2 π n$ , for any integer $n$